- FindStat

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000651
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000141
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000209
(load all 7 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the maximum of this value over all cycles in the permutation.

Matching statistic: St000503
(load all 2 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal difference between two elements in a common block.

Matching statistic: St000956
(load all 7 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal displacement of a permutation. This is

$\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation

$\pi$ of

$\{1,\ldots,n\}$ . This statistic without the absolute value is the maximal drop size [[St000141]].

Matching statistic: St000777
(load all 38 compositions to match this statistic)

Mapping

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1] => [2,1] => [1,2] => ([],2)
 => ? = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,3,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,2} + 1
[2,1,3] => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,2} + 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,1,2} + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,4,2] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 4 = 3 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[2,3,1,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,4,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,3,1] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,4,1] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => [4,2,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3} + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 5 = 4 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,4,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,5,2,3] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,4,5,3,2] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 5 = 4 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,5,3,4,2] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 5 = 4 + 1
[1,5,4,2,3] => [1,5,4,2,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,4,5,3] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,1,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,1,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,4,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,4,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,5,1,4] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,3,5,4,1] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,4,1,3,5] => [2,4,1,3,5] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,4,1,5,3] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,4,3,1,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,4,3,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,4,5,1,3] => [1,3,5,2,4] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,4,5,3,1] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,5,1,3,4] => [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,1,4,3] => [2,5,1,4,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,3,1,4] => [1,5,3,2,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,3,4,1] => [1,5,2,4,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 5 = 4 + 1
[2,5,4,1,3] => [2,5,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,5,4,3,1] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,4,2,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1,4,5,2] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,1,5,4,2] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,2,4,5,1] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,2,5,4,1] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,4,1,5,2] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,4,2,5,1] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,4,5,2,1] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,5,4,2,1] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,1,2,5,3] => [3,1,2,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,1,3,5,2] => [3,1,2,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,1,5,3,2] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,2,1,5,3] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,2,3,5,1] => [3,1,2,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,2,5,3,1] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St001232
(load all 30 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => [1,0]
 => [1,0]
 => 0
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ? = 2
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {1,2,2,3,3,3}
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3}
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3}
[1,4,2,3] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {1,2,2,3,3,3}
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {1,2,2,3,3,3}
[3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {1,2,2,3,3,3}
[4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[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]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[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]
 => 4
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5}

Description

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

Matching statistic: St000454
(load all 57 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => [1] => ([],1)
 => 0
[1,2] => ([],2)
 => [2] => ([],2)
 => 0
[2,1] => ([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => [3] => ([],3)
 => 0
[1,3,2] => ([(1,2)],3)
 => [1,2] => ([(1,2)],3)
 => 1
[2,1,3] => ([(1,2)],3)
 => [1,2] => ([(1,2)],3)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2
[1,2,3,4] => ([],4)
 => [4] => ([],4)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [1,3] => ([(2,3)],4)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [1,3] => ([(2,3)],4)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[2,1,3,4] => ([(2,3)],4)
 => [1,3] => ([(2,3)],4)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,2,3,3,3,3,3}
[1,2,3,4,5] => ([],5)
 => [5] => ([],5)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,3,2,4,5] => ([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,1,3,4,5] => ([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4}

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St001330
(load all 7 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,3,3,3} + 1
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,3,3,3} + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,3,3,3} + 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,3,3,3} + 1
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,3,3,3,3} + 1
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,5,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,5,4,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,2,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,4,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,5,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[3,5,2,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,3,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,2,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,2,4,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,3,2,4] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,3,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,4,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,1,4,3,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[5,2,1,3,4] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St001645
(load all 3 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,3,3,3,3} + 1
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4} + 1
[2,4,5,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,4,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,2,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1

Description

The pebbling number of a connected graph.

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St001879The 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. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2.