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Matching statistic: St000702
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 2
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 4
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 4
[1,4,3,5,2] => 3
[1,4,5,2,3] => 3
[1,4,5,3,2] => 3
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000105
(load all 168 compositions to match this statistic)
(load all 168 compositions to match this statistic)
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%
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> 2
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2] => {{1},{2,3}}
=> 2
[2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => {{1,2,3}}
=> 1
[3,1,2] => {{1,3},{2}}
=> 2
[3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> 3
[3,1,4,2] => {{1,3,4},{2}}
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> 3
[4,1,3,2] => {{1,4},{2},{3}}
=> 3
[4,2,1,3] => {{1,4},{2},{3}}
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> 3
[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}}
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 4
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 4
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 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: St000010
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
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%
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
[1,2] => {{1},{2}}
=> [1,1]
=> 2
[2,1] => {{1,2}}
=> [2]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> 3
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 3
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> 3
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 4
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 4
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> 3
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000507
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [[1,2]]
=> 2
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2
[3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[2,3,4,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[2,4,3,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 3
[3,1,4,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[3,2,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 3
[3,2,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,4,1,2] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[3,4,2,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,1,2,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 3
[4,1,3,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[4,2,1,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[4,2,3,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[4,3,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[4,3,2,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,2,4,5,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4
[1,2,5,4,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,3,4,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3
[1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 3
[1,3,5,4,2] => [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4
[1,4,2,5,3] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4
[1,4,3,5,2] => [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 3
[1,4,5,2,3] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 3
[1,4,5,3,2] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 3
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000211
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
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%
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
[1,2] => [2,1] => {{1,2}}
=> 1 = 2 - 1
[2,1] => [1,2] => {{1},{2}}
=> 0 = 1 - 1
[1,2,3] => [2,3,1] => {{1,2,3}}
=> 2 = 3 - 1
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> 1 = 2 - 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> 1 = 2 - 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 0 = 1 - 1
[3,1,2] => [3,1,2] => {{1,3},{2}}
=> 1 = 2 - 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1 = 2 - 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 3 = 4 - 1
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 3 - 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 2 = 3 - 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1 = 2 - 1
[1,4,2,3] => [2,4,1,3] => {{1,2,4},{3}}
=> 2 = 3 - 1
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 3 - 1
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 2 = 3 - 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1 = 2 - 1
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0 = 1 - 1
[2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 2 - 1
[3,1,2,4] => [3,4,2,1] => {{1,3},{2,4}}
=> 2 = 3 - 1
[3,1,4,2] => [3,1,2,4] => {{1,3},{2},{4}}
=> 1 = 2 - 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 3 - 1
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 2 - 1
[3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[3,4,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1 = 2 - 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 2 = 3 - 1
[4,1,3,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 2 = 3 - 1
[4,2,1,3] => [4,3,1,2] => {{1,4},{2,3}}
=> 2 = 3 - 1
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 3 - 1
[4,3,1,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 1 = 2 - 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1 = 2 - 1
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 4 = 5 - 1
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 3 = 4 - 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 3 = 4 - 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2 = 3 - 1
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 3 = 4 - 1
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3 = 4 - 1
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 3 = 4 - 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 2 = 3 - 1
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 2 = 3 - 1
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1 = 2 - 1
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,5},{3},{4}}
=> 2 = 3 - 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 3 - 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 3 = 4 - 1
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 2 = 3 - 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 3 = 4 - 1
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 3 - 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 2 = 3 - 1
[1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 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: St000093
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent 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%
Mp00071: Permutations —descent 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
[1,2] => [1,2] => [2] => ([],2)
=> 2
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 3
[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)
=> 1
[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)
=> 4
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 3
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[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)
=> 1
[2,4,1,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 3
[3,1,4,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> 3
[4,1,3,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[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)
=> 4
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4
[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)
=> 2
[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)
=> 4
[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)
=> 4
[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
[1,4,5,3,2] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(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
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to 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%
Mp00128: Set partitions —to 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
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 2
[2,1] => {{1,2}}
=> [2] => ([],2)
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> 1
[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)
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4] => ([],4)
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[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)
=> 5
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 2
[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)
=> 4
[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)
=> 4
[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
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(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
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to 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%
Mp00128: Set partitions —to 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
[1,2] => {{1},{2}}
=> [1,1] => ([(0,1)],2)
=> 2
[2,1] => {{1,2}}
=> [2] => ([],2)
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[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)
=> 1
[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)
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4] => ([],4)
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[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)
=> 5
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 4
[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)
=> 2
[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)
=> 4
[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)
=> 4
[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
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [1,2,2] => ([(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: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1]
=> [2]
=> 2
[2,1] => {{1,2}}
=> [2]
=> [1,1]
=> 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 3
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,1,1]
=> 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 3
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 2
[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},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 3
[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}}
=> [1,1,1,1,1]
=> [5]
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [3,2]
=> 3
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [3,2]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1] => 11 => 2
[2,1] => {{1,2}}
=> [2] => 10 => 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => 111 => 3
[1,3,2] => {{1},{2,3}}
=> [1,2] => 110 => 2
[2,1,3] => {{1,2},{3}}
=> [2,1] => 101 => 2
[2,3,1] => {{1,2,3}}
=> [3] => 100 => 1
[3,1,2] => {{1,3},{2}}
=> [2,1] => 101 => 2
[3,2,1] => {{1,3},{2}}
=> [2,1] => 101 => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 1110 => 3
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 1101 => 3
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 1100 => 2
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,2,1] => 1101 => 3
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 1101 => 3
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 1011 => 3
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 1010 => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 1001 => 2
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 1000 => 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1] => 1001 => 2
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1001 => 2
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 3
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => 1001 => 2
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1011 => 3
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 1001 => 2
[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},{2},{3}}
=> [2,1,1] => 1011 => 3
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 3
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 3
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1011 => 3
[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}}
=> [1,1,1,1,1] => 11111 => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 11110 => 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 11101 => 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 11100 => 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 11101 => 4
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 11101 => 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 11011 => 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 11010 => 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 11001 => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 11000 => 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 11001 => 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 4
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 11011 => 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 11001 => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 11010 => 3
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [1,2,2] => 11010 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000378The diagonal inversion number of an integer partition. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry 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. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000245The number of ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000167The number of leaves of an ordered tree. St000470The number of runs in a permutation. St000703The number of deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000822The Hadwiger number of the graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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.
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