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Your data matches 104 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000668
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St001280
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000319
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1 = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St001556
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 67%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[1,4,7],[2,5],[3,6]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[[1,6,7],[2],[3],[4],[5]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,5,7],[2],[3],[4],[6]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,4,7],[2],[3],[5],[6]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,3,7],[2],[4],[5],[6]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,2,7],[3],[4],[5],[6]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,5,6],[2],[3],[4],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,4,6],[2],[3],[5],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,3,6],[2],[4],[5],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,2,6],[3],[4],[5],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,4,5],[2],[3],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,3,5],[2],[4],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,2,5],[3],[4],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,3,4],[2],[5],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,2,4],[3],[5],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,6],[2,7],[3],[4],[5]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,5],[2,7],[3],[4],[6]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,4],[2,7],[3],[5],[6]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,3],[2,7],[4],[5],[6]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,2],[3,7],[4],[5],[6]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,5],[2,6],[3],[4],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,4],[2,6],[3],[5],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,3],[2,6],[4],[5],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,2],[3,6],[4],[5],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,4],[2,5],[3],[6],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,3],[2,5],[4],[6],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,2],[3,5],[4],[6],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,3],[2,4],[5],[6],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,7],[2],[3],[4],[5],[6]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,6],[2],[3],[4],[5],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,5],[2],[3],[4],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,4],[2],[3],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,3],[2],[4],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[[1,5,6,7,8],[2],[3],[4]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,4,6,7,8],[2],[3],[5]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,3,6,7,8],[2],[4],[5]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,2,6,7,8],[3],[4],[5]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,4,5,7,8],[2],[3],[6]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,3,5,7,8],[2],[4],[6]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[1,2,5,7,8],[3],[4],[6]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St000883
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,6],[3,4],[5],[7]]
=> [7,5,3,4,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 1
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001196
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001196: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001196: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 1
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,6,7,8],[2],[4],[5]]
=> [5,4,2,1,3,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,6,7,8],[3],[4],[5]]
=> [5,4,3,1,2,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,5,7,8],[2],[3],[6]]
=> [6,3,2,1,4,5,7,8] => ?
=> ?
=> ? = 1
[[1,3,5,7,8],[2],[4],[6]]
=> [6,4,2,1,3,5,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,5,7,8],[3],[4],[6]]
=> [6,4,3,1,2,5,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,4,7,8],[2],[5],[6]]
=> [6,5,2,1,3,4,7,8] => ?
=> ?
=> ? = 1
[[1,2,4,7,8],[3],[5],[6]]
=> [6,5,3,1,2,4,7,8] => ?
=> ?
=> ? = 1
[[1,2,3,7,8],[4],[5],[6]]
=> [6,5,4,1,2,3,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,5,6,8],[2],[3],[7]]
=> [7,3,2,1,4,5,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,5,6,8],[2],[4],[7]]
=> [7,4,2,1,3,5,6,8] => ?
=> ?
=> ? = 1
[[1,2,5,6,8],[3],[4],[7]]
=> [7,4,3,1,2,5,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,4,6,8],[2],[5],[7]]
=> [7,5,2,1,3,4,6,8] => ?
=> ?
=> ? = 1
[[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3,6,8],[4],[5],[7]]
=> [7,5,4,1,2,3,6,8] => ?
=> ?
=> ? = 1
[[1,3,4,5,8],[2],[6],[7]]
=> [7,6,2,1,3,4,5,8] => ?
=> ?
=> ? = 1
[[1,2,4,5,8],[3],[6],[7]]
=> [7,6,3,1,2,4,5,8] => ?
=> ?
=> ? = 1
[[1,2,3,5,8],[4],[6],[7]]
=> [7,6,4,1,2,3,5,8] => ?
=> ?
=> ? = 1
[[1,2,3,4,8],[5],[6],[7]]
=> [7,6,5,1,2,3,4,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,5,6,7],[2],[3],[8]]
=> [8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,5,6,7],[2],[4],[8]]
=> [8,4,2,1,3,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,5,6,7],[3],[4],[8]]
=> [8,4,3,1,2,5,6,7] => ?
=> ?
=> ? = 1
[[1,3,4,6,7],[2],[5],[8]]
=> [8,5,2,1,3,4,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,4,6,7],[3],[5],[8]]
=> [8,5,3,1,2,4,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3,6,7],[4],[5],[8]]
=> [8,5,4,1,2,3,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,4,5,7],[2],[6],[8]]
=> [8,6,2,1,3,4,5,7] => ?
=> ?
=> ? = 1
[[1,2,4,5,7],[3],[6],[8]]
=> [8,6,3,1,2,4,5,7] => ?
=> ?
=> ? = 1
[[1,2,3,5,7],[4],[6],[8]]
=> [8,6,4,1,2,3,5,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3,4,7],[5],[6],[8]]
=> [8,6,5,1,2,3,4,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,3,4,5,6],[2],[7],[8]]
=> [8,7,2,1,3,4,5,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,4,5,6],[3],[7],[8]]
=> [8,7,3,1,2,4,5,6] => ?
=> ?
=> ? = 1
[[1,2,3,5,6],[4],[7],[8]]
=> [8,7,4,1,2,3,5,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3,4,6],[5],[7],[8]]
=> [8,7,5,1,2,3,4,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,7,8],[2,5],[4,6]]
=> [4,6,2,5,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,2,7,8],[3,5],[4,6]]
=> [4,6,3,5,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,7,8],[2,4],[5,6]]
=> [5,6,2,4,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,4,6,8],[2,5],[3,7]]
=> [3,7,2,5,1,4,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,6,8],[2,5],[4,7]]
=> [4,7,2,5,1,3,6,8] => ?
=> ?
=> ? = 2
[[1,2,6,8],[3,5],[4,7]]
=> [4,7,3,5,1,2,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,6,8],[2,4],[5,7]]
=> [5,7,2,4,1,3,6,8] => ?
=> ?
=> ? = 2
[[1,2,6,8],[3,4],[5,7]]
=> [5,7,3,4,1,2,6,8] => ?
=> ?
=> ? = 2
[[1,4,5,8],[2,6],[3,7]]
=> [3,7,2,6,1,4,5,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,5,8],[2,6],[4,7]]
=> [4,7,2,6,1,3,5,8] => ?
=> ?
=> ? = 2
[[1,2,5,8],[3,6],[4,7]]
=> [4,7,3,6,1,2,5,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,3,4,8],[2,6],[5,7]]
=> [5,7,2,6,1,3,4,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[[1,2,4,8],[3,6],[5,7]]
=> [5,7,3,6,1,2,4,8] => ?
=> ?
=> ? = 2
Description
The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St000122
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000122: Binary trees ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 0 = 1 - 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,6,7,8],[2],[4],[5]]
=> [5,4,2,1,3,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,6,7,8],[3],[4],[5]]
=> [5,4,3,1,2,6,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,4,5,7,8],[2],[3],[6]]
=> [6,3,2,1,4,5,7,8] => ?
=> ?
=> ? = 1 - 1
[[1,3,5,7,8],[2],[4],[6]]
=> [6,4,2,1,3,5,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,5,7,8],[3],[4],[6]]
=> [6,4,3,1,2,5,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,4,7,8],[2],[5],[6]]
=> [6,5,2,1,3,4,7,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,4,7,8],[3],[5],[6]]
=> [6,5,3,1,2,4,7,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,3,7,8],[4],[5],[6]]
=> [6,5,4,1,2,3,7,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,4,5,6,8],[2],[3],[7]]
=> [7,3,2,1,4,5,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,5,6,8],[2],[4],[7]]
=> [7,4,2,1,3,5,6,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,5,6,8],[3],[4],[7]]
=> [7,4,3,1,2,5,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,4,6,8],[2],[5],[7]]
=> [7,5,2,1,3,4,6,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,3,6,8],[4],[5],[7]]
=> [7,5,4,1,2,3,6,8] => ?
=> ?
=> ? = 1 - 1
[[1,3,4,5,8],[2],[6],[7]]
=> [7,6,2,1,3,4,5,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,4,5,8],[3],[6],[7]]
=> [7,6,3,1,2,4,5,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,3,5,8],[4],[6],[7]]
=> [7,6,4,1,2,3,5,8] => ?
=> ?
=> ? = 1 - 1
[[1,2,3,4,8],[5],[6],[7]]
=> [7,6,5,1,2,3,4,8] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,4,5,6,7],[2],[3],[8]]
=> [8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,5,6,7],[2],[4],[8]]
=> [8,4,2,1,3,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,5,6,7],[3],[4],[8]]
=> [8,4,3,1,2,5,6,7] => ?
=> ?
=> ? = 1 - 1
[[1,3,4,6,7],[2],[5],[8]]
=> [8,5,2,1,3,4,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,4,6,7],[3],[5],[8]]
=> [8,5,3,1,2,4,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,3,6,7],[4],[5],[8]]
=> [8,5,4,1,2,3,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,4,5,7],[2],[6],[8]]
=> [8,6,2,1,3,4,5,7] => ?
=> ?
=> ? = 1 - 1
[[1,2,4,5,7],[3],[6],[8]]
=> [8,6,3,1,2,4,5,7] => ?
=> ?
=> ? = 1 - 1
[[1,2,3,5,7],[4],[6],[8]]
=> [8,6,4,1,2,3,5,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,3,4,7],[5],[6],[8]]
=> [8,6,5,1,2,3,4,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,3,4,5,6],[2],[7],[8]]
=> [8,7,2,1,3,4,5,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,4,5,6],[3],[7],[8]]
=> [8,7,3,1,2,4,5,6] => ?
=> ?
=> ? = 1 - 1
[[1,2,3,5,6],[4],[7],[8]]
=> [8,7,4,1,2,3,5,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,3,4,6],[5],[7],[8]]
=> [8,7,5,1,2,3,4,6] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 1 - 1
[[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,7,8],[2,5],[4,6]]
=> [4,6,2,5,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,2,7,8],[3,5],[4,6]]
=> [4,6,3,5,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,7,8],[2,4],[5,6]]
=> [5,6,2,4,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,4,6,8],[2,5],[3,7]]
=> [3,7,2,5,1,4,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,6,8],[2,5],[4,7]]
=> [4,7,2,5,1,3,6,8] => ?
=> ?
=> ? = 2 - 1
[[1,2,6,8],[3,5],[4,7]]
=> [4,7,3,5,1,2,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,6,8],[2,4],[5,7]]
=> [5,7,2,4,1,3,6,8] => ?
=> ?
=> ? = 2 - 1
[[1,2,6,8],[3,4],[5,7]]
=> [5,7,3,4,1,2,6,8] => ?
=> ?
=> ? = 2 - 1
[[1,4,5,8],[2,6],[3,7]]
=> [3,7,2,6,1,4,5,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,5,8],[2,6],[4,7]]
=> [4,7,2,6,1,3,5,8] => ?
=> ?
=> ? = 2 - 1
[[1,2,5,8],[3,6],[4,7]]
=> [4,7,3,6,1,2,5,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,3,4,8],[2,6],[5,7]]
=> [5,7,2,6,1,3,4,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 - 1
[[1,2,4,8],[3,6],[5,7]]
=> [5,7,3,6,1,2,4,8] => ?
=> ?
=> ? = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[[.,.],.]]]}}} in a binary tree.
[[oeis:A086581]] counts binary trees avoiding this pattern.
The following 94 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000054The first entry of the permutation. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000990The first ascent of a permutation. St000546The number of global descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000842The breadth of a permutation. St001513The number of nested exceedences of a permutation. St000487The length of the shortest cycle of a permutation. St000732The number of double deficiencies of a permutation. St000989The number of final rises of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001469The holeyness of a permutation. St001549The number of restricted non-inversions between exceedances. St000710The number of big deficiencies of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000210Minimum over maximum difference of elements in cycles. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001403The number of vertical separators in a permutation. St000654The first descent of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St000056The decomposition (or block) number of a permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000486The number of cycles of length at least 3 of a permutation. St000570The Edelman-Greene number of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001733The number of weak left to right maxima of a Dyck path. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000221The number of strong fixed points of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000274The number of perfect matchings of a graph. St000365The number of double ascents of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001386The number of prime labellings of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001434The number of negative sum pairs of a signed permutation. St000741The Colin de Verdière graph invariant. St001260The permanent of an alternating sign matrix. St000069The number of maximal elements of a poset. St000889The number of alternating sign matrices with the same antidiagonal sums.
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