- FindStat

Your data matches 98 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000834
(load all 16 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000834: 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] => [1,2] => 1
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,2,3] => 1
[2,1,3] => [1,2,3] => 1
[2,3,1] => [1,2,3] => 1
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,3,4] => 1
[1,3,2,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,2,3,4] => 1
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [1,2,3,4] => 1
[2,3,1,4] => [1,2,3,4] => 1
[2,3,4,1] => [1,2,3,4] => 1
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 2
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 2
[3,4,2,1] => [1,3,2,4] => 2
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [1,4,2,3] => 2
[4,3,2,1] => [1,4,2,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,3,4,5] => 1
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,2,3,4,5] => 1
[1,3,4,5,2] => [1,2,3,4,5] => 1
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => 2
[1,4,3,5,2] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => 2

Description

The number of right outer peaks of a permutation. A right outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$n$ if

$w_n > w_{n-1}$ . In other words, it is a peak in the word

$[w_1,..., w_n,0]$ .

Matching statistic: St000035
(load all 15 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000291
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ 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] => 10 => 1
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [1,2,3] => [3] => 100 => 1
[1,3,2] => [1,2,3] => [3] => 100 => 1
[2,1,3] => [1,2,3] => [3] => 100 => 1
[2,3,1] => [1,2,3] => [3] => 100 => 1
[3,1,2] => [1,3,2] => [2,1] => 101 => 1
[3,2,1] => [1,3,2] => [2,1] => 101 => 1
[1,2,3,4] => [1,2,3,4] => [4] => 1000 => 1
[1,2,4,3] => [1,2,3,4] => [4] => 1000 => 1
[1,3,2,4] => [1,2,3,4] => [4] => 1000 => 1
[1,3,4,2] => [1,2,3,4] => [4] => 1000 => 1
[1,4,2,3] => [1,2,4,3] => [3,1] => 1001 => 1
[1,4,3,2] => [1,2,4,3] => [3,1] => 1001 => 1
[2,1,3,4] => [1,2,3,4] => [4] => 1000 => 1
[2,1,4,3] => [1,2,3,4] => [4] => 1000 => 1
[2,3,1,4] => [1,2,3,4] => [4] => 1000 => 1
[2,3,4,1] => [1,2,3,4] => [4] => 1000 => 1
[2,4,1,3] => [1,2,4,3] => [3,1] => 1001 => 1
[2,4,3,1] => [1,2,4,3] => [3,1] => 1001 => 1
[3,1,2,4] => [1,3,2,4] => [2,2] => 1010 => 2
[3,1,4,2] => [1,3,4,2] => [3,1] => 1001 => 1
[3,2,1,4] => [1,3,2,4] => [2,2] => 1010 => 2
[3,2,4,1] => [1,3,4,2] => [3,1] => 1001 => 1
[3,4,1,2] => [1,3,2,4] => [2,2] => 1010 => 2
[3,4,2,1] => [1,3,2,4] => [2,2] => 1010 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 1
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => 1011 => 1
[4,2,3,1] => [1,4,2,3] => [2,2] => 1010 => 2
[4,3,1,2] => [1,4,2,3] => [2,2] => 1010 => 2
[4,3,2,1] => [1,4,2,3] => [2,2] => 1010 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => 10000 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => 10001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => 10001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => 10000 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => 10001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => 10001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => 10010 => 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => 10001 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => 10010 => 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => 10001 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => 10010 => 2

Description

The number of descents of a binary word.

Matching statistic: St000473

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000473: Integer partitions ⟶ 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] => [2]
 => 1
[2,1] => [[1],[2]]
 => [2] => [2]
 => 1
[1,2,3] => [[1,2,3]]
 => [3] => [3]
 => 1
[1,3,2] => [[1,2],[3]]
 => [3] => [3]
 => 1
[2,1,3] => [[1,3],[2]]
 => [2,1] => [2,1]
 => 1
[2,3,1] => [[1,2],[3]]
 => [3] => [3]
 => 1
[3,1,2] => [[1,3],[2]]
 => [2,1] => [2,1]
 => 1
[3,2,1] => [[1],[2],[3]]
 => [3] => [3]
 => 1
[1,2,3,4] => [[1,2,3,4]]
 => [4] => [4]
 => 1
[1,2,4,3] => [[1,2,3],[4]]
 => [4] => [4]
 => 1
[1,3,2,4] => [[1,2,4],[3]]
 => [3,1] => [3,1]
 => 1
[1,3,4,2] => [[1,2,3],[4]]
 => [4] => [4]
 => 1
[1,4,2,3] => [[1,2,4],[3]]
 => [3,1] => [3,1]
 => 1
[1,4,3,2] => [[1,2],[3],[4]]
 => [4] => [4]
 => 1
[2,1,3,4] => [[1,3,4],[2]]
 => [2,2] => [2,2]
 => 2
[2,1,4,3] => [[1,3],[2,4]]
 => [2,2] => [2,2]
 => 2
[2,3,1,4] => [[1,2,4],[3]]
 => [3,1] => [3,1]
 => 1
[2,3,4,1] => [[1,2,3],[4]]
 => [4] => [4]
 => 1
[2,4,1,3] => [[1,2],[3,4]]
 => [3,1] => [3,1]
 => 1
[2,4,3,1] => [[1,2],[3],[4]]
 => [4] => [4]
 => 1
[3,1,2,4] => [[1,3,4],[2]]
 => [2,2] => [2,2]
 => 2
[3,1,4,2] => [[1,3],[2,4]]
 => [2,2] => [2,2]
 => 2
[3,2,1,4] => [[1,4],[2],[3]]
 => [3,1] => [3,1]
 => 1
[3,2,4,1] => [[1,3],[2],[4]]
 => [2,2] => [2,2]
 => 2
[3,4,1,2] => [[1,2],[3,4]]
 => [3,1] => [3,1]
 => 1
[3,4,2,1] => [[1,2],[3],[4]]
 => [4] => [4]
 => 1
[4,1,2,3] => [[1,3,4],[2]]
 => [2,2] => [2,2]
 => 2
[4,1,3,2] => [[1,3],[2],[4]]
 => [2,2] => [2,2]
 => 2
[4,2,1,3] => [[1,4],[2],[3]]
 => [3,1] => [3,1]
 => 1
[4,2,3,1] => [[1,3],[2],[4]]
 => [2,2] => [2,2]
 => 2
[4,3,1,2] => [[1,4],[2],[3]]
 => [3,1] => [3,1]
 => 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => [4] => [4]
 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
 => [5] => [5]
 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => [5] => [5]
 => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [4,1] => [4,1]
 => 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => [5] => [5]
 => 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => [4,1] => [4,1]
 => 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [5] => [5]
 => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,2] => [3,2]
 => 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,2] => [3,2]
 => 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => [4,1] => [4,1]
 => 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => [5] => [5]
 => 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => [4,1] => [4,1]
 => 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => [5] => [5]
 => 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => [3,2] => [3,2]
 => 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => [3,2] => [3,2]
 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [4,1] => [4,1]
 => 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => [3,2] => [3,2]
 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [4,1] => [4,1]
 => 1

Description

The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].

Matching statistic: St000994
(load all 9 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000994: 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] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[1,3,2] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,1,3] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,1,2] => 1
[3,1,2] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [2,3,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,1,2,3] => 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [4,1,3,2] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [4,3,1,2] => 2
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [4,2,3,1] => 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => 2

Description

The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation

$\pi$ is an index

$i$ such that

$\pi^{-1}(i) < i > \pi(i)$ . Analogously, a '''cycle valley''' is an index

$i$ such that

$\pi^{-1}(i) > i < \pi(i)$ . Clearly, every cycle of

$\pi$ contains as many peaks as valleys.

Matching statistic: St001280
(load all 2 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ 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] => [2]
 => 1
[2,1] => [1,2] => [2] => [2]
 => 1
[1,2,3] => [1,2,3] => [3] => [3]
 => 1
[1,3,2] => [1,2,3] => [3] => [3]
 => 1
[2,1,3] => [1,2,3] => [3] => [3]
 => 1
[2,3,1] => [1,2,3] => [3] => [3]
 => 1
[3,1,2] => [1,3,2] => [2,1] => [2,1]
 => 1
[3,2,1] => [1,3,2] => [2,1] => [2,1]
 => 1
[1,2,3,4] => [1,2,3,4] => [4] => [4]
 => 1
[1,2,4,3] => [1,2,3,4] => [4] => [4]
 => 1
[1,3,2,4] => [1,2,3,4] => [4] => [4]
 => 1
[1,3,4,2] => [1,2,3,4] => [4] => [4]
 => 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [3,1]
 => 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [3,1]
 => 1
[2,1,3,4] => [1,2,3,4] => [4] => [4]
 => 1
[2,1,4,3] => [1,2,3,4] => [4] => [4]
 => 1
[2,3,1,4] => [1,2,3,4] => [4] => [4]
 => 1
[2,3,4,1] => [1,2,3,4] => [4] => [4]
 => 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [3,1]
 => 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [3,1]
 => 1
[3,1,2,4] => [1,3,2,4] => [2,2] => [2,2]
 => 2
[3,1,4,2] => [1,3,4,2] => [3,1] => [3,1]
 => 1
[3,2,1,4] => [1,3,2,4] => [2,2] => [2,2]
 => 2
[3,2,4,1] => [1,3,4,2] => [3,1] => [3,1]
 => 1
[3,4,1,2] => [1,3,2,4] => [2,2] => [2,2]
 => 2
[3,4,2,1] => [1,3,2,4] => [2,2] => [2,2]
 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [2,1,1]
 => 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [2,2]
 => 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [2,1,1]
 => 1
[4,2,3,1] => [1,4,2,3] => [2,2] => [2,2]
 => 2
[4,3,1,2] => [1,4,2,3] => [2,2] => [2,2]
 => 2
[4,3,2,1] => [1,4,2,3] => [2,2] => [2,2]
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [4,1]
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [4,1]
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [5]
 => 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [4,1]
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [4,1]
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [3,2]
 => 2
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [4,1]
 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [3,2]
 => 2
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [4,1]
 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [3,2]
 => 2

Description

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

Matching statistic: St000636

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000636: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The hull number of a graph. The convex hull of a set of vertices

$S$ of a graph is the smallest set

$h(S)$ such that for any pair

$u,v\in h(S)$ all vertices on a shortest path from

$u$ to

$v$ are also in

$h(S)$ . The hull number is the size of the smallest set

$S$ such that

$h(S)$ is the set of all vertices.

Matching statistic: St001654

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The monophonic hull number of a graph. The monophonic hull of a set of vertices

$M$ of a graph

$G$ is the set of vertices that lie on at least one induced path between vertices in

$M$ . The monophonic hull number is the size of the smallest set

$M$ such that the monophonic hull of

$M$ is all of

$G$ . For example, the monophonic hull number of a graph

$G$ with

$n$ vertices is

$n$ if and only if

$G$ is a disjoint union of complete graphs.

Matching statistic: St001655

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001655: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The general position number of a graph. A set

$S$ of vertices in a graph

$G$ is a general position set if no three vertices of

$S$ lie on a shortest path between any two of them.

Matching statistic: St001656

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The monophonic position number of a graph. A subset

$M$ of the vertex set of a graph is a monophonic position set if no three vertices of

$M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.

The following 88 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001883The mutual visibility number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000353The number of inner valleys of a permutation. St000201The number of leaf nodes in a binary tree. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000884The number of isolated descents of a permutation. St001729The number of visible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001928The number of non-overlapping descents in a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000659The number of rises of length at least 2 of a Dyck path. St000808The number of up steps of the associated bargraph. St001487The number of inner corners of a skew partition. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000354The number of recoils of a permutation. St000386The number of factors DDU in a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St001298The number of repeated entries in the Lehmer code of a permutation. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000628The balance of a binary word. St000711The number of big exceedences of a permutation. St000668The least common multiple of the parts of the partition. St000260The radius of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000455The second largest eigenvalue of a graph if it is integral. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001890The maximum magnitude of the Möbius function of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001946The number of descents in a parking function. St000093The cardinality of a maximal independent set of vertices of a graph. St000679The pruning number of an ordered tree. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001857The number of edges in the reduced word graph of a signed permutation. St001624The breadth of a lattice. St001271The competition number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian.