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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000566
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[5,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,4,1]
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[4,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [2,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [2,1]
=> 1
[7,2,1]
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[6,4]
=> [4]
=> [2,2]
=> [2]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[5,4,1]
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[5,3,2]
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [3,2]
=> [2]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000017
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 35%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 35%
Values
[2,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,2]
=> [2,2]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[5,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,4]
=> [4]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[3,3,2]
=> [3,2]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 1
[6,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[5,4]
=> [4]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,4,1]
=> [4,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[4,3,2]
=> [3,2]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 1
[7,2,1]
=> [2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[6,4]
=> [4]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[6,2,1,1]
=> [2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[5,4,1]
=> [4,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[5,3,2]
=> [3,2]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[5,3,1,1]
=> [3,1,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> ? = 1
[3,3,3,3]
=> [3,3,3]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 3
[3,3,3,2,1]
=> [3,3,2,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1
[3,3,2,2,2]
=> [3,2,2,2]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 3
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 4
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [4,2,1,1,1,1]
=> [[1,6,9,10],[2,8],[3],[4],[5],[7]]
=> ? = 1
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> ? = 0
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 1
[4,4,4,1]
=> [4,4,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[4,4,3,2]
=> [4,3,2]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 2
[4,4,3,1,1]
=> [4,3,1,1]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 3
[4,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 2
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 2
[4,3,3,3]
=> [3,3,3]
=> [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> ? = 3
[4,3,3,2,1]
=> [3,3,2,1]
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1
[4,3,2,2,2]
=> [3,2,2,2]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 3
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 4
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1
[3,3,3,3,1]
=> [3,3,3,1]
=> [6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> ? = 3
[3,3,3,2,2]
=> [3,3,2,2]
=> [6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> ? = 1
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> ? = 6
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> ? = 3
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 4
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
Matching statistic: St000782
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 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]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> ? = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> ? = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1 + 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> ? = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> ? = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> ? = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> ? = 1 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1 + 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 0 + 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1 + 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0 + 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 0 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 0 + 1
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 2 + 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? = 0 + 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> ? = 0 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> ? = 0 + 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 0 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
=> ? = 1 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[14,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[13,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001491
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[2,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [4]
=> 10000 => ? = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [4]
=> 10000 => ? = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => ? = 1 + 1
[5,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[4,4]
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 1 + 1
[4,2,2]
=> [2,2]
=> [4]
=> 10000 => ? = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [5]
=> 100000 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> 100010 => ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2,2]
=> 11100 => ? = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,1,1]
=> 1000110 => ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,2]
=> 100100 => ? = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> 1001010 => ? = 1 + 1
[6,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[5,4]
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 1 + 1
[5,2,2]
=> [2,2]
=> [4]
=> 10000 => ? = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[4,4,1]
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 1 + 1
[4,3,2]
=> [3,2]
=> [5]
=> 100000 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [4,1]
=> 100010 => ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => ? = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [5,1]
=> 1000010 => ? = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [3,3]
=> 11000 => ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [2,2,2]
=> 11100 => ? = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [4,1,1]
=> 1000110 => ? = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,2]
=> 100100 => ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [4,1,1,1]
=> 10001110 => ? = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [4,3]
=> 101000 => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [3,3,1]
=> 110010 => ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> 1001010 => ? = 1 + 1
[7,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[6,4]
=> [4]
=> [1,1,1,1]
=> 11110 => ? = 1 + 1
[6,2,2]
=> [2,2]
=> [4]
=> 10000 => ? = 0 + 1
[6,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[5,4,1]
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 1 + 1
[5,3,2]
=> [3,2]
=> [5]
=> 100000 => ? = 0 + 1
[5,3,1,1]
=> [3,1,1]
=> [4,1]
=> 100010 => ? = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 11010 => ? = 0 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 100110 => ? = 0 + 1
[4,4,2]
=> [4,2]
=> [2,2,1,1]
=> 110110 => ? = 1 + 1
[4,4,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => ? = 2 + 1
[4,3,2,1]
=> [3,2,1]
=> [5,1]
=> 1000010 => ? = 0 + 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [3,3]
=> 11000 => ? = 1 + 1
[4,2,2,2]
=> [2,2,2]
=> [2,2,2]
=> 11100 => ? = 0 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [4,1,1]
=> 1000110 => ? = 0 + 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [4,2]
=> 100100 => ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 101010 => ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [2,2,2,1]
=> 111010 => ? = 0 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [5,2]
=> 1000100 => ? = 1 + 1
[8,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[7,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[8,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[9,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[10,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[11,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[12,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
[14,2,1]
=> [2,1]
=> [3]
=> 1000 => 1 = 0 + 1
[13,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1100 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001722
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 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]
=> 110101010100 => ? = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1 + 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => ? = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => ? = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 1 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1 + 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 0 + 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 + 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 0 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 0 + 1
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 + 1
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 + 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 0 + 1
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 0 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 0 + 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 0 + 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 101110010100 => ? = 1 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[14,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[13,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000075
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000075: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0 + 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0 + 3
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 0 + 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 + 3
[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]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> ? = 1 + 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 1 + 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0 + 3
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0 + 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 1 + 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 0 + 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 + 3
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 0 + 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 0 + 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> ? = 1 + 3
[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]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? = 1 + 3
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 1 + 3
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0 + 3
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 1 + 3
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0 + 3
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 1 + 3
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 0 + 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 + 3
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1 + 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 0 + 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 0 + 3
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> ? = 1 + 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[1,2,3,4,8],[5,6,7,9,10]]
=> ? = 0 + 3
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,2,3,6,8,10],[4,5,7,9,11,12]]
=> ? = 0 + 3
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
=> ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
=> ? = 1 + 3
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 1 + 3
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 0 + 3
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 1 + 3
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 0 + 3
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> ? = 1 + 3
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> ? = 0 + 3
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> ? = 0 + 3
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 1 + 3
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 2 + 3
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 0 + 3
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1 + 3
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> ? = 0 + 3
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> ? = 0 + 3
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> ? = 1 + 3
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 3
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,3,4,5,8,10],[2,6,7,9,11,12]]
=> ? = 1 + 3
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
[14,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 0 + 3
[13,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 3 = 0 + 3
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St001583
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 3
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 0 + 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 3
[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]
=> [7,6,1,2,3,4,5] => ? = 1 + 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 3
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 0 + 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 3
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 0 + 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 0 + 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0 + 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 1 + 3
[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]
=> [8,7,1,2,3,4,5,6] => ? = 1 + 3
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 3
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 3
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 3
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 3
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 3
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 0 + 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 3
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 0 + 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 0 + 3
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0 + 3
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 1 + 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 0 + 3
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ? = 0 + 3
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 0 + 3
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => ? = 1 + 3
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 3
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 0 + 3
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 3
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 0 + 3
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1 + 3
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 0 + 3
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 0 + 3
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1 + 3
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 2 + 3
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 0 + 3
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1 + 3
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 0 + 3
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 0 + 3
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 0 + 3
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 1 + 3
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 3
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 1 + 3
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
[14,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 0 + 3
[13,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 0 + 3
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St000689
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[2,2,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,2]
=> [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
[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
[4,2,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[3,2,2]
=> [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
[3,2,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [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
[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]
=> ? = 0
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,2,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[4,4]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[4,2,2]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[4,2,1,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[3,3,2]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,3,1,1]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[3,2,2,1]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[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]
=> ? = 0
[2,2,2,2]
=> [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]
=> ? = 0
[2,2,2,1,1]
=> [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]
=> ? = 0
[2,2,1,1,1,1]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[2,1,1,1,1,1,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [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]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[6,2,1]
=> [5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[5,4]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[5,2,2]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[5,2,1,1]
=> [4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[4,4,1]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[4,3,2]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,3,1,1]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[4,2,2,1]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[4,2,1,1,1]
=> [5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 0
[3,3,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[3,3,1,1,1]
=> [5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[3,2,2,2]
=> [4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[3,2,2,1,1]
=> [5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 0
[3,2,1,1,1,1]
=> [6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0
[3,1,1,1,1,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,2,2,2,1]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0
[2,2,2,1,1,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1,1]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,2,1]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[6,4]
=> [5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[6,2,2]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[6,2,1,1]
=> [5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[5,4,1]
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 1
[5,3,2]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,3,1,1]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[5,2,2,1]
=> [4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 0
[5,2,1,1,1]
=> [5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[4,4,2]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[4,4,1,1]
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,3,1,1,1]
=> [5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[4,2,2,2]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 0
[4,2,2,1,1]
=> [5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[4,2,1,1,1,1]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[4,1,1,1,1,1,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[3,3,2,2]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 0
[3,3,2,1,1]
=> [5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[3,3,1,1,1,1]
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[3,2,2,2,1]
=> [5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0
[3,2,2,1,1,1]
=> [6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0
[3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[3,1,1,1,1,1,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,2,2,2,2]
=> [5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 0
[2,2,2,2,1,1]
=> [6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 0
[2,2,2,1,1,1,1]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[8,2,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0
[7,4]
=> [6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[7,2,2]
=> [6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 0
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].
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