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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001175
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Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: 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
St001175: 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]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 0 = 1 - 1
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000345
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: 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
St000345: 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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[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]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: 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
St000935: 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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[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]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St001389
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: 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
St001389: 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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[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]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[[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]
=> 1
[[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]
=> 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[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]
=> 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [2]
=> 2
[[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]
=> 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001033
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001596
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0 = 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],[]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1],[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],[1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 2 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,1]
=> [[3,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0 = 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]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[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]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[2],[2],[2],[1],[1],[1],[1]]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 2 - 1
[[2],[2],[2],[2],[1],[1]]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3 - 1
[[2],[2],[2],[2],[2]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 4 - 1
[[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]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 2 - 1
[[1,1],[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 3 - 1
[[1,1],[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 4 - 1
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St000017
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 75%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 75%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2],[3]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> 0 = 1 - 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]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,2,3],[4],[5],[6]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,3,4],[5],[6]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> 0 = 1 - 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]]
=> 0 = 1 - 1
[[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]]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,2,3,4],[5,6],[7]]
=> 1 = 2 - 1
[[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]]
=> 0 = 1 - 1
[[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]]
=> 0 = 1 - 1
[[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]]
=> 1 = 2 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,2,3,4],[5],[6],[7]]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 0 = 1 - 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]]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,3,4,5],[6],[7]]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,3,4,5],[6,7]]
=> 1 = 2 - 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]]
=> 0 = 1 - 1
[[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],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,3,4,5,6],[7]]
=> 0 = 1 - 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]]
=> 0 = 1 - 1
[[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]]
=> 0 = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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],[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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 - 1
[[3],[3],[3]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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 - 1
[[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 - 1
[[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]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 1
[[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]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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],[1,1,1]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,3,4,5,6],[7,8,9]]
=> ? = 3 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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 - 1
[[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]]
=> ? = 1 - 1
[[4],[3],[2]]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,4,5,6,7],[8,9]]
=> ? = 2 - 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 - 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]]
=> ? = 1 - 1
[[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]]
=> ? = 1 - 1
[[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 - 1
[[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]]
=> ? = 1 - 1
[[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 - 1
[[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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 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]]
=> ? = 1 - 1
[[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]]
=> ? = 2 - 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]]
=> ? = 2 - 1
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St001556
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[[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]
=> [2,3,4,5,6,7,1] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[[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]
=> [2,3,4,5,6,7,8,1] => ? = 1
[[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]
=> [3,2,4,5,6,7,1] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 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]
=> [3,2,4,5,6,7,1] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => ? = 1
[[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]
=> [2,3,4,5,6,7,8,9,1] => ? = 1
[[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]
=> [3,2,4,5,6,7,8,1] => ? = 1
[[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]
=> [3,4,2,5,6,7,1] => ? = 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ? = 3
[[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]
=> [3,2,4,5,6,7,8,1] => ? = 1
[[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]
=> [3,4,2,5,6,7,1] => ? = 1
[[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]
=> [3,4,5,2,6,1] => ? = 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]
=> [3,4,5,6,1,2] => ? = 3
[[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]
=> [4,2,3,5,6,7,1] => ? = 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => ? = 1
[[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]
=> [4,2,3,5,6,7,1] => ? = 1
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => ? = 1
[[2,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => ? = 1
[[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]
=> [4,2,3,5,6,7,1] => ? = 1
[[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]
=> [4,3,2,5,6,1] => ? = 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => ? = 1
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => ? = 1
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => ? = 1
[[2,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => ? = 1
[[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]
=> [5,2,3,4,6,1] => ? = 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[5],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[4,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 1
[[4,1],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[3,2],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 1
[[3,2],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => ? = 1
[[3,1,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? = 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: St001232
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 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]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[[4],[2],[2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3],[3],[3]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1],[2,1],[2,1]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,1],[1,1,1],[1,1,1]]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[5],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[4,1],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[4,1],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,2],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,2],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,1,1],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[3,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2,1],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,2,1],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1,1],[2],[2]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[2,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1,1,1],[1,1],[1,1]]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[[4],[3],[3]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[3,1],[3],[3]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[3,1],[2,1],[2,1]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,2],[2,1],[2,1]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1,1],[2,1],[2,1]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[2,1,1],[1,1,1],[1,1,1]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
[[1,1,1,1],[1,1,1],[1,1,1]]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 3 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000314
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 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] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 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]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 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] => ? = 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] => ? = 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] => ? = 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]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 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] => ? = 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
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 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] => ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[[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]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 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] => ? = 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] => ? = 2
[[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
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[[4],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 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] => ? = 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 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]]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 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
[[5],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 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]]
=> [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],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 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] => ? = 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] => ? = 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] => ? = 2
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 3
[[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] => ? = 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] => ? = 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] => ? = 2
[[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] => ? = 3
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
The following 1 statistic also match your data. Click on any of them to see the details.
St000654The first descent of a permutation.
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