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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St001250
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: 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
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001804
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 2
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> 3
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2],[3],[4],[5],[6]]
=> 4
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 3
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 5
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,2,3],[4],[5],[6],[7]]
=> 4
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,2,3,4],[5,6],[7]]
=> 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,2,3],[4],[5],[6],[7]]
=> 4
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,2,3,4],[5,6],[7]]
=> 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 2
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 2
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 7
[[2],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 6
[[2],[2],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 5
[[2],[2],[2],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 4
[[2],[2],[2],[2],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 3
[[1,1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 6
[[1,1],[1,1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 5
[[1,1],[1,1],[1,1],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 4
[[1,1],[1,1],[1,1],[1,1],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 3
[[3],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 5
[[3],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[3],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> ? = 3
[[3],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 2
[[3],[3],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[3],[3],[2],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[3],[3],[3]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> ? = 0
[[2,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 5
[[2,1],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[2,1],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> ? = 3
[[2,1],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 2
[[2,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[2,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> ? = 3
[[2,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 2
[[2,1],[2,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[2,1],[2,1],[2],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[2,1],[2,1],[1,1],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[2,1],[2,1],[2,1]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> ? = 0
[[1,1,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 5
[[1,1,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[1,1,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> ? = 3
[[1,1,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> ? = 2
[[1,1,1],[1,1,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[1,1,1],[1,1,1],[1,1],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[1,1,1],[1,1,1],[1,1,1]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> ? = 0
[[4],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[4],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[4],[2],[2],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[4],[3],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 2
[[4],[3],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 1
[[4],[4],[1]]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,2,3,4,5,6,7,8],[9]]
=> ? = 1
[[3,1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> ? = 4
[[3,1],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[3,1],[2],[2],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[3,1],[1,1],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> ? = 3
[[3,1],[1,1],[1,1],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> ? = 2
[[3,1],[3],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 2
[[3,1],[3],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 1
[[3,1],[2,1],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> ? = 2
[[3,1],[2,1],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 1
[[3,1],[2,1],[1,1]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 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.
Matching statistic: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 1 + 2
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 2 + 2
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 3 = 1 + 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 3 + 2
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 4 = 2 + 2
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 3 = 1 + 2
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 4 = 2 + 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 3 = 1 + 2
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 1 + 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 1 + 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 6 = 4 + 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 5 = 3 + 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 4 = 2 + 2
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 3 = 1 + 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 5 = 3 + 2
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 4 = 2 + 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 3 = 1 + 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 4 = 2 + 2
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 3 = 1 + 2
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 4 = 2 + 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 3 = 1 + 2
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 3 = 1 + 2
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 4 = 2 + 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 3 = 1 + 2
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 1 + 2
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 1 + 2
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 1 + 2
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 1 + 2
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 7 = 5 + 2
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 6 = 4 + 2
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 5 = 3 + 2
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 4 = 2 + 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 6 = 4 + 2
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 5 = 3 + 2
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 4 = 2 + 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 5 = 3 + 2
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 4 = 2 + 2
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 3 = 1 + 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> 3 = 1 + 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 5 = 3 + 2
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 4 = 2 + 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 3 = 1 + 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 4 = 2 + 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 3 = 1 + 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> 3 = 1 + 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 5 = 3 + 2
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 4 = 2 + 2
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,2,3,4,5,6,7,8,9]]
=> ? = 7 + 2
[[2],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 6 + 2
[[2],[2],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> ? = 5 + 2
[[2],[2],[2],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> ? = 4 + 2
[[2],[2],[2],[2],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 3 + 2
[[1,1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 6 + 2
[[1,1],[1,1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> ? = 5 + 2
[[1,1],[1,1],[1,1],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> ? = 4 + 2
[[1,1],[1,1],[1,1],[1,1],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 3 + 2
[[3],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 5 + 2
[[3],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 4 + 2
[[3],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 3 + 2
[[3],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 2 + 2
[[3],[3],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> ? = 3 + 2
[[3],[3],[2],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 2 + 2
[[3],[3],[3]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 0 + 2
[[2,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 5 + 2
[[2,1],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 4 + 2
[[2,1],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 3 + 2
[[2,1],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 2 + 2
[[2,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 4 + 2
[[2,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 3 + 2
[[2,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 2 + 2
[[2,1],[2,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> ? = 3 + 2
[[2,1],[2,1],[2],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 2 + 2
[[2,1],[2,1],[1,1],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 2 + 2
[[2,1],[2,1],[2,1]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 0 + 2
[[1,1,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 5 + 2
[[1,1,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 4 + 2
[[1,1,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 3 + 2
[[1,1,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 2 + 2
[[1,1,1],[1,1,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> ? = 3 + 2
[[1,1,1],[1,1,1],[1,1],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 2 + 2
[[1,1,1],[1,1,1],[1,1,1]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 0 + 2
[[4],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 4 + 2
[[4],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> ? = 3 + 2
[[4],[2],[2],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> ? = 2 + 2
[[4],[3],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> ? = 2 + 2
[[4],[3],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 1 + 2
[[4],[4],[1]]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> ? = 1 + 2
[[3,1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 4 + 2
[[3,1],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> ? = 3 + 2
[[3,1],[2],[2],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> ? = 2 + 2
[[3,1],[1,1],[1],[1],[1]]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> ? = 3 + 2
[[3,1],[1,1],[1,1],[1]]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> ? = 2 + 2
[[3,1],[3],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> ? = 2 + 2
[[3,1],[3],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 1 + 2
[[3,1],[2,1],[1],[1]]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> ? = 2 + 2
[[3,1],[2,1],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 1 + 2
[[3,1],[2,1],[1,1]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 1 + 2
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St001227
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 22%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 22%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[[1],[1],[1],[1]]
=> [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 = 2 + 2
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1]]
=> [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]
=> ? = 3 + 2
[[2],[1],[1],[1]]
=> [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 = 2 + 2
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[[1,1],[1],[1],[1]]
=> [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 = 2 + 2
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 + 2
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 2
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 1 + 2
[[3],[1],[1],[1]]
=> [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]
=> 4 = 2 + 2
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[2,1],[1],[1],[1]]
=> [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]
=> 4 = 2 + 2
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[1,1,1],[1],[1],[1]]
=> [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]
=> 4 = 2 + 2
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[[4],[1],[1]]
=> [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]
=> 3 = 1 + 2
[[3,1],[1],[1]]
=> [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]
=> 3 = 1 + 2
[[2,2],[1],[1]]
=> [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]
=> 3 = 1 + 2
[[2,1,1],[1],[1]]
=> [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]
=> 3 = 1 + 2
[[1,1,1,1],[1],[1]]
=> [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]
=> 3 = 1 + 2
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 5 + 2
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4 + 2
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 2
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 4 = 2 + 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[[3],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 2
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 1 + 2
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 1 + 2
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[[4],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4 = 2 + 2
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 1 + 2
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 2
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 6 + 2
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 5 + 2
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 + 2
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 5 + 2
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 + 2
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 2
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 4 + 2
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[[2,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 4 + 2
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[[2,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[[1,1,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 4 + 2
[[1,1,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 2
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 2
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 2
[[2,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 2
[[1,1,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 2
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[[5],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[[4,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
[[4,1],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[[3,2],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
[[3,2],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 2
[[3,1,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 2
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001232
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 56%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 4 + 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[3],[2],[2],[1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[3],[3],[1],[1]]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[[3],[3],[2]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[[2,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 4 + 1
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[[2,1],[2],[2],[1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000021
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000541
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000619
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 3 = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 2 = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000831
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 44%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2 = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3 = 2 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2 = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 4 = 3 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 3 = 2 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 5 = 4 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 4 = 3 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 3 = 2 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 4 = 3 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 3 = 2 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2 = 1 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 3 = 2 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2 = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 3 = 2 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2 = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2 = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 3 = 2 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2 = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 2 = 1 + 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 5 + 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 4 + 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 2 + 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 1 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 1 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 1 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 3 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 2 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? = 1 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 2 + 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 1 + 1
[[5],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 + 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 6 + 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => ? = 5 + 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ? = 4 + 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ? = 3 + 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => ? = 5 + 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ? = 4 + 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2 + 1
Description
The number of indices that are either descents or recoils.
This is, for a permutation $\pi$ of length $n$, this statistics counts the set
$$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
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