- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000673
(load all 56 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => 0
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [3,2,1] => 2
[3,1,2] => [2,3,1] => 3
[3,2,1] => [3,1,2] => 3
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,3,4,2] => 3
[1,4,3,2] => [1,4,2,3] => 3
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => 2
[2,4,1,3] => [3,2,4,1] => 3
[2,4,3,1] => [4,2,1,3] => 3
[3,1,2,4] => [2,3,1,4] => 3
[3,1,4,2] => [3,4,1,2] => 4
[3,2,1,4] => [3,1,2,4] => 3
[3,2,4,1] => [4,3,2,1] => 4
[3,4,1,2] => [2,4,3,1] => 3
[3,4,2,1] => [4,1,3,2] => 3
[4,1,2,3] => [2,3,4,1] => 4
[4,1,3,2] => [3,4,2,1] => 4
[4,2,1,3] => [3,1,4,2] => 4
[4,2,3,1] => [4,3,1,2] => 4
[4,3,1,2] => [2,4,1,3] => 4
[4,3,2,1] => [4,1,2,3] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,4,5,3] => 3
[1,2,5,4,3] => [1,2,5,3,4] => 3
[1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,4,3,5,2] => 3
[1,3,5,4,2] => [1,5,3,2,4] => 3
[1,4,2,3,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,4,5,2,3] => 4
[1,4,3,2,5] => [1,4,2,3,5] => 3
[1,4,3,5,2] => [1,5,4,3,2] => 4
[1,4,5,2,3] => [1,3,5,4,2] => 3
[1,4,5,3,2] => [1,5,2,4,3] => 3

Description

The number of non-fixed points of a permutation. In other words, this statistic is

$n$ minus the number of fixed points ([[St000022]]) of

$\pi$ .

Matching statistic: St000235
(load all 22 compositions to match this statistic)

Mapping

Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 2
[2,1,3] => [2,1,3] => [1,3,2] => 2
[2,3,1] => [3,2,1] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 3
[3,2,1] => [2,3,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 2
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 2
[1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 3
[1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 3
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 4
[2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 2
[2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 2
[2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [3,2,4,1] => [2,1,3,4] => 3
[3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 3
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => 4
[3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 3
[3,2,4,1] => [4,3,2,1] => [3,2,1,4] => 4
[3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 3
[3,4,2,1] => [2,4,3,1] => [1,3,2,4] => 3
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 4
[4,1,3,2] => [4,3,1,2] => [4,2,1,3] => 4
[4,2,1,3] => [2,4,1,3] => [1,4,2,3] => 4
[4,2,3,1] => [3,4,2,1] => [3,1,2,4] => 4
[4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 4
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 2
[1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => 3
[1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 4
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [5,3,4,2,1] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => 3
[1,3,5,4,2] => [1,4,3,5,2] => [5,3,2,4,1] => 3
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => 3
[1,4,2,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => 4
[1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 3
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 4
[1,4,5,2,3] => [1,5,2,4,3] => [3,5,4,2,1] => 3
[1,4,5,3,2] => [1,3,5,4,2] => [5,2,4,3,1] => 3

Description

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index

$i < n$ such that

$\pi_i = i+1$ , or the index

$i = n$ if

$\pi_n = 1$ .

Matching statistic: St001279
(load all 4 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,1]
 => 0
[2,1] => [2,1] => [2]
 => 2
[1,2,3] => [1,2,3] => [1,1,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => 2
[2,1,3] => [2,1,3] => [2,1]
 => 2
[2,3,1] => [3,2,1] => [2,1]
 => 2
[3,1,2] => [2,3,1] => [3]
 => 3
[3,2,1] => [3,1,2] => [3]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => 2
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => 2
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => 2
[1,4,2,3] => [1,3,4,2] => [3,1]
 => 3
[1,4,3,2] => [1,4,2,3] => [3,1]
 => 3
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,1,4,3] => [2,2]
 => 4
[2,3,1,4] => [3,2,1,4] => [2,1,1]
 => 2
[2,3,4,1] => [4,2,3,1] => [2,1,1]
 => 2
[2,4,1,3] => [3,2,4,1] => [3,1]
 => 3
[2,4,3,1] => [4,2,1,3] => [3,1]
 => 3
[3,1,2,4] => [2,3,1,4] => [3,1]
 => 3
[3,1,4,2] => [3,4,1,2] => [2,2]
 => 4
[3,2,1,4] => [3,1,2,4] => [3,1]
 => 3
[3,2,4,1] => [4,3,2,1] => [2,2]
 => 4
[3,4,1,2] => [2,4,3,1] => [3,1]
 => 3
[3,4,2,1] => [4,1,3,2] => [3,1]
 => 3
[4,1,2,3] => [2,3,4,1] => [4]
 => 4
[4,1,3,2] => [3,4,2,1] => [4]
 => 4
[4,2,1,3] => [3,1,4,2] => [4]
 => 4
[4,2,3,1] => [4,3,1,2] => [4]
 => 4
[4,3,1,2] => [2,4,1,3] => [4]
 => 4
[4,3,2,1] => [4,1,2,3] => [4]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => 2
[1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
 => 2
[1,2,5,3,4] => [1,2,4,5,3] => [3,1,1]
 => 3
[1,2,5,4,3] => [1,2,5,3,4] => [3,1,1]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => 4
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
 => 2
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
 => 2
[1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
 => 3
[1,3,5,4,2] => [1,5,3,2,4] => [3,1,1]
 => 3
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,1]
 => 3
[1,4,2,5,3] => [1,4,5,2,3] => [2,2,1]
 => 4
[1,4,3,2,5] => [1,4,2,3,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
 => 4
[1,4,5,2,3] => [1,3,5,4,2] => [3,1,1]
 => 3
[1,4,5,3,2] => [1,5,2,4,3] => [3,1,1]
 => 3

Description

The sum of the parts of an integer partition that are at least two.

Matching statistic: St000896
(load all 10 compositions to match this statistic)

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [1,2] => [[1,0],[0,1]]
 => 0
[2,1] => [2,1] => [2,1] => [[0,1],[1,0]]
 => 2
[1,2,3] => [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 0
[1,3,2] => [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 2
[2,1,3] => [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2
[2,3,1] => [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 2
[3,1,2] => [3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 3
[3,2,1] => [3,2,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => 3
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 3
[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 3
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 4
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 4
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 3
[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 3
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 4
[4,2,3,1] => [4,1,3,2] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
 => 4
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 2
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 3
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 4
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 3
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 3

Description

The number of zeros on the main diagonal of an alternating sign matrix.

Matching statistic: St001005

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[2,1] => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 2
[1,2,3] => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 2
[2,1,3] => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 2
[2,3,1] => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 2
[3,1,2] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 3
[3,2,1] => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 3

Description

The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.

Matching statistic: St001182

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 5 = 4 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5 = 4 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5 = 4 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 4 = 3 + 1

Description

Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St001232

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%

Values

[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 0
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? = 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? = 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ? = 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 4
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? = 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001879
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%

Values

[1,2] => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ? = 0
[2,1] => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(2,1)],3)
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 0
[1,3,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[2,1,3] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2
[2,3,1] => [1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 2
[3,1,2] => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ? = 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001880
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 67%

Values

[1,2] => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,1] => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 4 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 2 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 4 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ? = 2 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 4 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 4 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4 + 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 5 + 1

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.