searching the database
Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000566
Mp00037: Graphs —to partition of connected components⟶ 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%
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
([],3)
=> [1,1,1]
=> [1,1]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> 3
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: St000185
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000185: 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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000185: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 3
Description
The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$
\sum_i \binom{\lambda^{\prime}_i}{2}
$$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St000017
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1
([],7)
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 1
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 3
([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
=> ? = 0
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? = 0
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 0
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> [6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> ? = 0
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> [6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 0
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ? = 0
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 0
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
=> ? = 0
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> [7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> ? = 0
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 0
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
=> ? = 0
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? = 0
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
=> ? = 0
([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> [8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> ? = 0
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? = 0
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? = 0
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
=> ? = 0
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 0
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? = 0
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? = 0
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? = 0
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 0
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
=> ? = 0
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? = 0
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2],[3]]
=> ? = 0
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> [7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> ? = 0
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? = 0
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? = 0
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? = 0
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? = 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: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 60%
Values
([],3)
=> []
=> []
=> []
=> ? = 0
([],4)
=> []
=> []
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([],5)
=> []
=> []
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(2,4),(3,4)],5)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([],6)
=> []
=> []
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(3,5),(4,5)],6)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
([],7)
=> []
=> []
=> []
=> ? = 0
([(5,6)],7)
=> [1]
=> [1,0]
=> [1,0]
=> 0
([(4,6),(5,6)],7)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([(3,6),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(3,6),(4,5)],7)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,3),(4,6),(5,6)],7)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(4,5),(4,6),(5,6)],7)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 3
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
([(0,1),(0,2),(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 3
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ?
=> ? = 0
([],8)
=> ?
=> ?
=> ?
=> ? = 0
([(4,7),(5,6)],8)
=> ?
=> ?
=> ?
=> ? = 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001559
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001559: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 80%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001559: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 80%
Values
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
([],7)
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(1,6),(2,5),(3,4)],7)
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 3
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 1
Description
The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions.
This statistic is the difference between [[St001558]] and [[St000018]].
A permutation is '''smooth''' if and only if this number is zero. Equivalently, this number is zero if and only if the permutation avoids the two patterns $4231$ and $3412$.
Matching statistic: St001845
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([],1)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([],5)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 1
([],6)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ?
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ?
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ?
=> ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ?
=> ? = 1
([],7)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(3,6),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
([(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
=> ? = 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ? = 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? = 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],6)
=> ?
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 0
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001613
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 1 + 1
([],6)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ?
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ?
=> ? = 3 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ?
=> ? = 3 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ?
=> ? = 1 + 1
([],7)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 3 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
=> ? = 0 + 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],6)
=> ?
=> ? = 2 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 3 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
Description
The binary logarithm of the size of the center of a lattice.
An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001719
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 1 + 1
([],6)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ?
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ?
=> ? = 3 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ?
=> ? = 3 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ?
=> ? = 1 + 1
([],7)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 3 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
=> ? = 0 + 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],6)
=> ?
=> ? = 2 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 3 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001881
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],4)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([],5)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ?
=> ? = 1 + 1
([],6)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ?
=> ? = 2 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ?
=> ? = 3 + 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ?
=> ? = 1 + 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 0 + 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ?
=> ? = 3 + 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ?
=> ? = 1 + 1
([],7)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(3,6),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1 + 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 3 + 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 0 + 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3 + 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
=> ? = 0 + 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 1
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],6)
=> ?
=> ? = 2 + 1
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1 + 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ?
=> ? = 0 + 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 1 + 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ?
=> ? = 1 + 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 0 + 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 0 + 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
=> ? = 0 + 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? = 1 + 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 3 + 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3 + 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 1 = 0 + 1
Description
The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St001677
Values
([],3)
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([],4)
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(0,3),(1,2)],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([],5)
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ?
=> ? = 1
([],6)
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(2,5),(3,4)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
([(1,2),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([],6)
=> ([],6)
=> ?
=> ? = 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,4),(3,5)],6)
=> ?
=> ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([],5)
=> ([],5)
=> ?
=> ? = 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],5)
=> ([],5)
=> ?
=> ? = 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],6)
=> ([],6)
=> ?
=> ? = 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> ([],6)
=> ?
=> ? = 1
([],7)
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(3,6),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
([(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? = 0
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
([(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ?
=> ? = 2
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ?
=> ? = 0
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
=> ?
=> ? = 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([],6)
=> ([],6)
=> ?
=> ? = 2
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ?
=> ? = 0
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> ?
=> ? = 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> ([(3,4),(3,5)],6)
=> ?
=> ? = 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 0
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 0
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,5),(3,6),(3,13),(4,12),(5,9),(5,11),(6,9),(6,10),(7,15),(8,15),(9,14),(10,7),(10,14),(11,8),(11,14),(12,7),(12,8),(13,10),(13,11),(13,12),(14,15)],16)
=> ? = 0
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 0
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ?
=> ? = 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ? = 3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 0
Description
The number of non-degenerate subsets of a lattice whose meet is the bottom element.
A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000934The 2-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001271The competition number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!