- FindStat

Your data matches 84 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000470
(load all 33 compositions to match this statistic)

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St000211
(load all 168 compositions to match this statistic)

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.

Matching statistic: St000105
(load all 28 compositions to match this statistic)

Mapping

Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]]

$S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of

$\{ 1,\ldots,n\}$ into

$k$ blocks, see [1].

Matching statistic: St000157
(load all 4 compositions to match this statistic)

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000010
(load all 6 compositions to match this statistic)

Mapping

Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000011

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ 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] => [2] => [1,1,0,0]
 => 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
 => 2
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 2
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 2
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 2
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,2,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000093

Mapping

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number

$\alpha(G)$ of

$G$ .

Matching statistic: St000097

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000098

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000167

Mapping

Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaves of an ordered tree. This is the number of nodes which do not have any children.

The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000507The number of ascents of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000354The number of recoils of a permutation. St000703The number of deficiencies of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000306The bounce count of a Dyck path. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000702The number of weak deficiencies of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000155The number of exceedances (also excedences) of a permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001812The biclique partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St001769The reflection length of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001935The number of ascents in a parking function.