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Matching statistic: St000208
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: 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
St000208: 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
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
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]]
=> ? = 2 - 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]]
=> ? = 2 - 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.
Matching statistic: St000654
(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
St000654: 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
St000654: 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 first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
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