searching the database
Your data matches 21 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: St000783
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => [3,1,1]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [3,1,1]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [3,1,1]
=> 2
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St001432
Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,3,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,1,2] => [1,2,3] => [1,2,3] => [3]
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,4,2,3] => [3,1,1]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,4,2,3] => [3,1,1]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [3,1,1]
=> 2
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001431
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001555
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 2
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => ? = 2
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 2
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ? = 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ? = 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ? = 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 3
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => ? = 2
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 3
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => ? = 2
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => ? = 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [1,5,3,6,2,4] => [1,5,3,6,2,4] => ? = 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,5,2,3,6] => ? = 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? = 3
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => ? = 3
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 3
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 3
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [1,4,6,2,5,3] => [1,4,6,2,5,3] => ? = 2
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => ? = 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 2
Description
The order of a signed permutation.
Matching statistic: St001553
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [3,4,1,2,5,6] => [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,6,5,4,2] => [1,3,6,2,4,5] => [6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
Matching statistic: St001195
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 1 - 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 1 - 1
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> ? = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,6,1,2,4,5] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [4,5,1,2,3,6] => [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,2,6,4,5,3] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,3,4] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,2,6,5,4,3] => [1,2,6,3,4,5] => [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [2,5,6,1,3,4] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [6,2,4,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,2,4,5,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,3,6,4,5,2] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,3,6,5,2,4] => [1,3,6,2,4,5] => [2,4,5,6,1,3] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St000298
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,4,5,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,4,5,3,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,1,3,4] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,5,1,4,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,5,3,1,4] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,4,1,3] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,2,4,5] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,2,5,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,1,4,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[3,1,4,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> ? = 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,1,4,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,1,4] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 2
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,3,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,3,5,2,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,3,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,2,5,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,2,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[1,4,5,2,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,4,5,3,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[1,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,2,4] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[1,5,3,4,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,2,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,3,5,4] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,1,4,5,3] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,1,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,1,5,4,3] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,1,4,5] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[2,3,1,5,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,1,5] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[2,4,1,3,5] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[2,4,1,5,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[2,4,3,1,5] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001491
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1] => => ? = 1 - 1
[1,2] => [1,2] => [1,2] => 0 => ? = 1 - 1
[2,1] => [1,2] => [1,2] => 0 => ? = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 00 => ? = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 10 => 1 = 2 - 1
[2,1,3] => [1,3,2] => [3,1,2] => 10 => 1 = 2 - 1
[2,3,1] => [1,2,3] => [1,2,3] => 00 => ? = 1 - 1
[3,1,2] => [1,2,3] => [1,2,3] => 00 => ? = 1 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 00 => ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 100 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,4,2] => [1,3,4,2] => [3,1,2,4] => 100 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[2,1,3,4] => [1,3,4,2] => [3,1,2,4] => 100 => 1 = 2 - 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1 = 2 - 1
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 100 => 1 = 2 - 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1 = 2 - 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => 1000 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => 0100 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 0100 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 1100 => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,2,4,5] => 1000 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,2,5,3] => [4,1,5,2,3] => 1100 => 1 = 2 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1 = 2 - 1
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => 1100 => 1 = 2 - 1
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [3,1,2,4,5] => 1000 => 1 = 2 - 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,5,1,2,4] => 1000 => 1 = 2 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[2,1,4,5,3] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1 = 2 - 1
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[2,3,1,4,5] => [1,4,5,2,3] => [4,5,1,2,3] => 1000 => 1 = 2 - 1
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 00000 => ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 10000 => ? = 2 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,5,2,3,4,6] => 01000 => ? = 2 - 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [5,1,2,3,4,6] => 10000 => ? = 2 - 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 01000 => ? = 2 - 1
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [1,6,2,3,4,5] => 01000 => ? = 2 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 00100 => ? = 2 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,1,6,2,3,5] => 11000 => ? = 2 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [1,4,2,3,5,6] => 01000 => ? = 2 - 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [4,1,2,3,5,6] => 10000 => ? = 2 - 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => 10000 => ? = 2 - 1
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,6,1,2,3,5] => 10000 => ? = 2 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 00100 => ? = 2 - 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [5,1,6,2,3,4] => 11000 => ? = 2 - 1
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [5,1,6,2,3,4] => 11000 => ? = 2 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => 00100 => ? = 2 - 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => 10000 => ? = 2 - 1
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [5,6,1,2,3,4] => 10000 => ? = 2 - 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => 00100 => ? = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St000640The rank of the largest boolean interval in a poset. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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!