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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001571
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: 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
St001571: 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]
=> 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]]
=> [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]
=> 1
[[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 Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St000657
(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
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [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]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[[2],[2],[1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => ? = 1
[[2],[2],[2],[1],[1],[1],[1]]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [2,2,2,1,1,1,1] => ? = 1
[[2],[2],[2],[2],[1],[1]]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
[[2],[2],[2],[2],[2]]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[[1,1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[[1,1],[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => ? = 1
[[1,1],[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [2,2,2,1,1,1,1] => ? = 1
[[1,1],[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [2,2,2,2,1,1] => ? = 1
[[1,1],[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 2
[[3],[1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 1
[[3],[2],[1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? = 1
[[3],[2],[2],[1],[1],[1]]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[[3],[2],[2],[2],[1]]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[[3],[3],[1],[1],[1],[1]]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => ? = 1
[[3],[3],[2],[1],[1]]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[[3],[3],[2],[2]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[[3],[3],[3],[1]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 2
[[2,1],[1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 1
[[2,1],[2],[1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? = 1
[[2,1],[2],[2],[1],[1],[1]]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[[2,1],[2],[2],[2],[1]]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[[2,1],[1,1],[1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? = 1
[[2,1],[1,1],[1,1],[1],[1],[1]]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[[2,1],[1,1],[1,1],[1,1],[1]]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[[2,1],[2,1],[1],[1],[1],[1]]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => ? = 1
[[2,1],[2,1],[2],[1],[1]]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[[2,1],[2,1],[2],[2]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[[2,1],[2,1],[1,1],[1],[1]]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[[2,1],[2,1],[1,1],[1,1]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[[2,1],[2,1],[2,1],[1]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 2
[[1,1,1],[1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 1
[[1,1,1],[1,1],[1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [3,2,1,1,1,1,1] => ? = 1
[[1,1,1],[1,1],[1,1],[1],[1],[1]]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [3,2,2,1,1,1] => ? = 1
[[1,1,1],[1,1],[1,1],[1,1],[1]]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [3,2,2,2,1] => ? = 1
[[1,1,1],[1,1,1],[1],[1],[1],[1]]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [3,3,1,1,1,1] => ? = 1
[[1,1,1],[1,1,1],[1,1],[1],[1]]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [3,3,2,1,1] => ? = 1
[[1,1,1],[1,1,1],[1,1],[1,1]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [3,3,2,2] => ? = 2
[[1,1,1],[1,1,1],[1,1,1],[1]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [3,3,3,1] => ? = 2
[[4],[1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [4,1,1,1,1,1,1] => ? = 1
[[4],[2],[1],[1],[1],[1]]
=> [4,2,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10]]
=> [4,2,1,1,1,1] => ? = 1
[[4],[2],[2],[1],[1]]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [4,2,2,1,1] => ? = 1
[[4],[2],[2],[2]]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> [4,2,2,2] => ? = 2
[[4],[3],[1],[1],[1]]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [4,3,1,1,1] => ? = 1
[[4],[3],[2],[1]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [4,3,2,1] => ? = 1
[[4],[3],[3]]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 3
[[4],[4],[1],[1]]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> [4,4,1,1] => ? = 1
[[4],[4],[2]]
=> [4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10]]
=> [4,4,2] => ? = 2
[[3,1],[1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10]]
=> [4,1,1,1,1,1,1] => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St000121
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000121: Binary trees ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000121: Binary trees ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 0 = 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]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[[1,1],[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]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 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]
=> [.,[[[[[.,.],.],.],.],.]]
=> 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]
=> [[.,.],[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 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]
=> [[.,.],[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> 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]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> 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]
=> [.,[[[[.,[.,.]],.],.],.]]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 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]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> 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]
=> [.,[[[[.,[.,.]],.],.],.]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 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]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> 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]
=> [[.,.],[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1 - 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[2,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[1,1,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 - 1
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 - 1
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 - 1
[[2,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 - 1
[[1,1,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[2,2,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[2,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[1,1,1,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 - 1
[[6],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[5,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[4,2],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[4,1,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[3,3],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[3,2,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[3,1,1,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[2,2,2],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[2,2,1,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[2,1,1,1,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[1,1,1,1,1,1],[1],[1]]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 - 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ? = 1 - 1
[[2],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1 - 1
[[2],[2],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> ? = 1 - 1
[[3],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1 - 1
[[3],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> ? = 1 - 1
[[2,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1 - 1
[[2,1],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> ? = 1 - 1
[[2,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> ? = 1 - 1
[[1,1,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> ? = 1 - 1
[[4],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[4],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> ? = 1 - 1
[[3,1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[3,1],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> ? = 1 - 1
[[3,1],[1,1],[1],[1],[1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> ? = 1 - 1
[[2,2],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
[[2,2],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> ? = 1 - 1
[[2,2],[1,1],[1],[1],[1]]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> ? = 1 - 1
[[2,1,1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,.]]]]}}} in a binary tree.
[[oeis:A036765]] counts binary trees avoiding this pattern.
Matching statistic: St000383
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ? = 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[[2,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[[2,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[[1,1,1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ? = 1
[[1,1,1],[1,1],[1],[1],[1]]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ? = 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[[4],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[4],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[[3,1],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[3,1],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[3,1],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[3,1],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[[2,2],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[2,2],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[2,2],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[2,2],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[2,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[[2,1,1],[2],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[2,1,1],[2],[2]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[2,1,1],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[2,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[1,1,1,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ? = 1
[[1,1,1,1],[1,1],[1],[1]]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ? = 1
[[1,1,1,1],[1,1],[1,1]]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ? = 2
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => ? = 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]]
=> [2,1,1,1,1,1,1,1] => ? = 1
[[2],[2],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [2,2,1,1,1,1,1] => ? = 1
[[2],[2],[2],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [2,2,2,1,1,1] => ? = 1
[[2],[2],[2],[2],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ? = 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]]
=> [2,1,1,1,1,1,1,1] => ? = 1
[[1,1],[1,1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [2,2,1,1,1,1,1] => ? = 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]]
=> [2,2,2,1,1,1] => ? = 1
[[1,1],[1,1],[1,1],[1,1],[1]]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ? = 1
[[3],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [3,1,1,1,1,1,1] => ? = 1
[[3],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [3,2,1,1,1,1] => ? = 1
[[3],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [3,2,2,1,1] => ? = 1
[[3],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ? = 2
[[3],[3],[1],[1],[1]]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [3,3,1,1,1] => ? = 1
[[3],[3],[2],[1]]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ? = 1
[[3],[3],[3]]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ? = 3
[[2,1],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [3,1,1,1,1,1,1] => ? = 1
[[2,1],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [3,2,1,1,1,1] => ? = 1
[[2,1],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [3,2,2,1,1] => ? = 1
[[2,1],[2],[2],[2]]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ? = 2
Description
The last part of an integer composition.
Matching statistic: St001803
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 0 = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 0 = 1 - 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0 = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0 = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1,1]
=> [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]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[[2],[2],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[[2],[2],[2],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[[2],[2],[2],[2],[1]]
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 1 - 1
[[1,1],[1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1,1]
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1],[1],[1]]
=> [2,2,2,1,1,1]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1,1],[1]]
=> [2,2,2,2,1]
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 1 - 1
[[3],[1],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[[3],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[[3],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[[3],[2],[2],[2]]
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 2 - 1
[[3],[3],[1],[1],[1]]
=> [3,3,1,1,1]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[[3],[3],[2],[1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 1 - 1
[[3],[3],[3]]
=> [3,3,3]
=> [3,3,3]
=> [[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]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[[2,1],[2],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[[2,1],[2],[2],[1],[1]]
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[[2,1],[2],[2],[2]]
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 2 - 1
[[2,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[[2,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[[2,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 2 - 1
[[2,1],[2,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[[2,1],[2,1],[2],[1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 1 - 1
[[2,1],[2,1],[1,1],[1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 1 - 1
[[2,1],[2,1],[2,1]]
=> [3,3,3]
=> [3,3,3]
=> [[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]
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1],[1],[1],[1]]
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1,1],[1],[1]]
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
[[1,1,1],[1,1],[1,1],[1,1]]
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 2 - 1
[[1,1,1],[1,1,1],[1],[1],[1]]
=> [3,3,1,1,1]
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> ? = 1 - 1
[[1,1,1],[1,1,1],[1,1],[1]]
=> [3,3,2,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 1 - 1
[[1,1,1],[1,1,1],[1,1,1]]
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 3 - 1
[[4],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 1 - 1
[[4],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[[4],[2],[2],[1]]
=> [4,2,2,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 1 - 1
[[4],[3],[1],[1]]
=> [4,3,1,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 1 - 1
[[4],[3],[2]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 - 1
[[4],[4],[1]]
=> [4,4,1]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
[[3,1],[1],[1],[1],[1],[1]]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 1 - 1
[[3,1],[2],[1],[1],[1]]
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[[3,1],[2],[2],[1]]
=> [4,2,2,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 1 - 1
[[3,1],[1,1],[1],[1],[1]]
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
[[3,1],[1,1],[1,1],[1]]
=> [4,2,2,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 1 - 1
[[3,1],[3],[1],[1]]
=> [4,3,1,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 1 - 1
[[3,1],[3],[2]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 - 1
[[3,1],[2,1],[1],[1]]
=> [4,3,1,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 1 - 1
[[3,1],[2,1],[2]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 - 1
[[3,1],[2,1],[1,1]]
=> [4,3,2]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 2 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St000654
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 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]
=> [3,1,4,5,6,7,8,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 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]
=> [8,1,4,5,6,7,2,3] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[[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]
=> [8,1,4,5,6,7,2,3] => ? = 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]
=> [6,1,4,5,2,7,3] => ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 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]
=> [3,1,7,5,6,2,4] => ? = 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 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]
=> [3,1,4,5,6,7,8,9,10,2] => ? = 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]
=> [9,1,4,5,6,7,8,2,3] => ? = 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]
=> [7,1,4,5,6,2,8,3] => ? = 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ? = 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => ? = 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]
=> [7,1,4,5,6,2,8,3] => ? = 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]
=> [5,1,4,2,6,7,3] => ? = 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 2
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? = 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 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]
=> [3,1,8,5,6,7,2,4] => ? = 1
[[2,1],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ? = 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]
=> [7,1,6,5,2,3,4] => ? = 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]
=> [3,1,8,5,6,7,2,4] => ? = 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]
=> [7,1,6,5,2,3,4] => ? = 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[[3,1],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 1
[[2,2],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 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]
=> [3,1,4,7,6,2,5] => ? = 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]
=> [3,1,4,7,6,2,5] => ? = 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1
[[5],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1
[[4,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[[4,1],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1
[[3,2],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[[3,2],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1
[[3,1,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 1
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 $\pi(n+1)=0$.
Matching statistic: St000689
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 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]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 - 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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1 - 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 1 - 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[5],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001217
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 67%
Values
[[1],[1],[1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[2],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[[2],[2],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[3],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 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]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[[2],[2],[2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[[3],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[2],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[4],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[3,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[2,2],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 - 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[3],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[3],[3],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 - 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[2],[2]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[2,1],[2,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 0 = 1 - 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]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 - 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[4],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[3,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[2,2],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[5],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 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]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 - 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 - 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[[3],[1],[1],[1],[1],[1]]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 - 1
[[3],[2],[1],[1],[1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 - 1
[[4],[1],[1],[1],[1]]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 - 1
[[5],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[5],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[4,1],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,2],[1,1],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[1],[1],[1]]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 - 1
[[3,1,1],[2],[1]]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 1 - 1
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000090
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Values
[[1],[1],[1]]
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1
[[1],[1],[1],[1]]
=> [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1
[[2],[1],[1]]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1
[[1,1],[1],[1]]
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1
[[1],[1],[1],[1],[1]]
=> [1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => 1
[[2],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => 1
[[2],[2],[1]]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1
[[1,1],[1],[1],[1]]
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => 1
[[1,1],[1,1],[1]]
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1
[[3],[1],[1]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => 1
[[2,1],[1],[1]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => 1
[[1,1,1],[1],[1]]
=> [3,1,1]
=> 100110 => [1,3,1,2] => 1
[[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => ? = 1
[[2],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 1
[[2],[2],[1],[1]]
=> [2,2,1,1]
=> 110110 => [1,1,2,1,2] => 1
[[2],[2],[2]]
=> [2,2,2]
=> 11100 => [1,1,1,3] => 2
[[1,1],[1],[1],[1],[1]]
=> [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 1
[[1,1],[1,1],[1],[1]]
=> [2,2,1,1]
=> 110110 => [1,1,2,1,2] => 1
[[1,1],[1,1],[1,1]]
=> [2,2,2]
=> 11100 => [1,1,1,3] => 2
[[3],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 1
[[3],[2],[1]]
=> [3,2,1]
=> 101010 => [1,2,2,2] => 1
[[2,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 1
[[2,1],[2],[1]]
=> [3,2,1]
=> 101010 => [1,2,2,2] => 1
[[2,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => [1,2,2,2] => 1
[[1,1,1],[1],[1],[1]]
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 1
[[1,1,1],[1,1],[1]]
=> [3,2,1]
=> 101010 => [1,2,2,2] => 1
[[4],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1
[[3,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1
[[2,2],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1
[[2,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1
[[1,1,1,1],[1],[1]]
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 1
[[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => ? = 1
[[2],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 1
[[2],[2],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => ? = 1
[[2],[2],[2],[1]]
=> [2,2,2,1]
=> 111010 => [1,1,1,2,2] => 1
[[1,1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 1
[[1,1],[1,1],[1],[1],[1]]
=> [2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => ? = 1
[[1,1],[1,1],[1,1],[1]]
=> [2,2,2,1]
=> 111010 => [1,1,1,2,2] => 1
[[3],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ? = 1
[[3],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ? = 1
[[3],[2],[2]]
=> [3,2,2]
=> 101100 => [1,2,1,3] => 2
[[3],[3],[1]]
=> [3,3,1]
=> 110010 => [1,1,3,2] => 1
[[2,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ? = 1
[[2,1],[2],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ? = 1
[[2,1],[2],[2]]
=> [3,2,2]
=> 101100 => [1,2,1,3] => 2
[[2,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ? = 1
[[2,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => [1,2,1,3] => 2
[[2,1],[2,1],[1]]
=> [3,3,1]
=> 110010 => [1,1,3,2] => 1
[[1,1,1],[1],[1],[1],[1]]
=> [3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ? = 1
[[1,1,1],[1,1],[1],[1]]
=> [3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ? = 1
[[1,1,1],[1,1],[1,1]]
=> [3,2,2]
=> 101100 => [1,2,1,3] => 2
[[1,1,1],[1,1,1],[1]]
=> [3,3,1]
=> 110010 => [1,1,3,2] => 1
[[4],[1],[1],[1]]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1
[[4],[2],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[3,1],[1],[1],[1]]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1
[[3,1],[2],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[3,1],[1,1],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[2,2],[1],[1],[1]]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1
[[2,2],[2],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[2,2],[1,1],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[2,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1
[[2,1,1],[2],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[2,1,1],[1,1],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[1,1,1,1],[1],[1],[1]]
=> [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 1
[[1,1,1,1],[1,1],[1]]
=> [4,2,1]
=> 1001010 => [1,3,2,2] => ? = 1
[[5],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[4,1],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[3,2],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[3,1,1],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[2,2,1],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[2,1,1,1],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[1,1,1,1,1],[1],[1]]
=> [5,1,1]
=> 10000110 => [1,5,1,2] => ? = 1
[[1],[1],[1],[1],[1],[1],[1],[1]]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => [1,1,1,1,1,1,1,1,2] => ? = 1
[[2],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => ? = 1
[[2],[2],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => ? = 1
[[2],[2],[2],[1],[1]]
=> [2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => ? = 1
[[2],[2],[2],[2]]
=> [2,2,2,2]
=> 111100 => [1,1,1,1,3] => 2
[[1,1],[1],[1],[1],[1],[1],[1]]
=> [2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => ? = 1
[[1,1],[1,1],[1],[1],[1],[1]]
=> [2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => ? = 1
[[1,1],[1,1],[1,1],[1],[1]]
=> [2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => ? = 1
[[1,1],[1,1],[1,1],[1,1]]
=> [2,2,2,2]
=> 111100 => [1,1,1,1,3] => 2
[[3],[3],[2]]
=> [3,3,2]
=> 110100 => [1,1,2,3] => 2
[[2,1],[2,1],[2]]
=> [3,3,2]
=> 110100 => [1,1,2,3] => 2
[[2,1],[2,1],[1,1]]
=> [3,3,2]
=> 110100 => [1,1,2,3] => 2
[[1,1,1],[1,1,1],[1,1]]
=> [3,3,2]
=> 110100 => [1,1,2,3] => 2
[[3],[3],[3]]
=> [3,3,3]
=> 111000 => [1,1,1,4] => 3
[[2,1],[2,1],[2,1]]
=> [3,3,3]
=> 111000 => [1,1,1,4] => 3
[[1,1,1],[1,1,1],[1,1,1]]
=> [3,3,3]
=> 111000 => [1,1,1,4] => 3
Description
The variation of a composition.
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: 67%
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: 67%
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]
=> ? = 1 + 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]
=> ? = 1 + 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.
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St000314The number of left-to-right-maxima of a permutation.
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