Your data matches 93 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 6 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 0 = 1 - 1
[1,2] => [2]
 => 0 = 1 - 1
[2,1] => [1,1]
 => 1 = 2 - 1
[1,2,3] => [3]
 => 0 = 1 - 1
[1,3,2] => [2,1]
 => 1 = 2 - 1
[2,1,3] => [2,1]
 => 1 = 2 - 1
[2,3,1] => [2,1]
 => 1 = 2 - 1
[3,1,2] => [2,1]
 => 1 = 2 - 1
[3,2,1] => [1,1,1]
 => 2 = 3 - 1
[1,2,3,4] => [4]
 => 0 = 1 - 1
[1,2,4,3] => [3,1]
 => 1 = 2 - 1
[1,3,2,4] => [3,1]
 => 1 = 2 - 1
[1,3,4,2] => [3,1]
 => 1 = 2 - 1
[1,4,2,3] => [3,1]
 => 1 = 2 - 1
[1,4,3,2] => [2,1,1]
 => 2 = 3 - 1
[2,1,3,4] => [3,1]
 => 1 = 2 - 1
[2,1,4,3] => [2,2]
 => 2 = 3 - 1
[2,3,1,4] => [3,1]
 => 1 = 2 - 1
[2,3,4,1] => [3,1]
 => 1 = 2 - 1
[2,4,1,3] => [2,2]
 => 2 = 3 - 1
[2,4,3,1] => [2,1,1]
 => 2 = 3 - 1
[3,1,2,4] => [3,1]
 => 1 = 2 - 1
[3,1,4,2] => [2,2]
 => 2 = 3 - 1
[3,2,1,4] => [2,1,1]
 => 2 = 3 - 1
[3,2,4,1] => [2,1,1]
 => 2 = 3 - 1
[3,4,1,2] => [2,2]
 => 2 = 3 - 1
[3,4,2,1] => [2,1,1]
 => 2 = 3 - 1
[4,1,2,3] => [3,1]
 => 1 = 2 - 1
[4,1,3,2] => [2,1,1]
 => 2 = 3 - 1
[4,2,1,3] => [2,1,1]
 => 2 = 3 - 1
[4,2,3,1] => [2,1,1]
 => 2 = 3 - 1
[4,3,1,2] => [2,1,1]
 => 2 = 3 - 1
[4,3,2,1] => [1,1,1,1]
 => 3 = 4 - 1
[1,2,3,4,5] => [5]
 => 0 = 1 - 1
[1,2,3,5,4] => [4,1]
 => 1 = 2 - 1
[1,2,4,3,5] => [4,1]
 => 1 = 2 - 1
[1,2,4,5,3] => [4,1]
 => 1 = 2 - 1
[1,2,5,3,4] => [4,1]
 => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1]
 => 2 = 3 - 1
[1,3,2,4,5] => [4,1]
 => 1 = 2 - 1
[1,3,2,5,4] => [3,2]
 => 2 = 3 - 1
[1,3,4,2,5] => [4,1]
 => 1 = 2 - 1
[1,3,4,5,2] => [4,1]
 => 1 = 2 - 1
[1,3,5,2,4] => [3,2]
 => 2 = 3 - 1
[1,3,5,4,2] => [3,1,1]
 => 2 = 3 - 1
[1,4,2,3,5] => [4,1]
 => 1 = 2 - 1
[1,4,2,5,3] => [3,2]
 => 2 = 3 - 1
[1,4,3,2,5] => [3,1,1]
 => 2 = 3 - 1
[1,4,3,5,2] => [3,1,1]
 => 2 = 3 - 1
[1,4,5,2,3] => [3,2]
 => 2 = 3 - 1
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000507
(load all 6 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => 1
[1,2] => [2]
 => [[1,2]]
 => 2
[2,1] => [1,1]
 => [[1],[2]]
 => 1
[1,2,3] => [3]
 => [[1,2,3]]
 => 3
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2
[2,3,1] => [2,1]
 => [[1,2],[3]]
 => 2
[3,1,2] => [2,1]
 => [[1,2],[3]]
 => 2
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => 1
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => 4
[1,2,4,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,3,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[2,1,3,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 3
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,3,4,1] => [3,1]
 => [[1,2,3],[4]]
 => 3
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => 3
[2,4,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => 3
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,2,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
[3,4,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,1,2,3] => [3,1]
 => [[1,2,3],[4]]
 => 3
[4,1,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,3,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => 5
[1,2,3,5,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,4,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,5,3,4] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,2,5,4,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,3,2,4,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,2,5,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 4
[1,3,4,2,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,3,5,2,4] => [3,2]
 => [[1,2,3],[4,5]]
 => 4
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,2,3,5] => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,4,2,5,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4
[1,4,3,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 3
[1,4,5,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 4
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: St000738
(load all 2 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => 1
[1,2] => [1,1]
 => [[1],[2]]
 => 2
[2,1] => [2]
 => [[1,2]]
 => 1
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => 2
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => 2
[2,3,1] => [2,1]
 => [[1,3],[2]]
 => 2
[3,1,2] => [2,1]
 => [[1,3],[2]]
 => 2
[3,2,1] => [3]
 => [[1,2,3]]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[1,3,4,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[1,4,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[1,4,3,2] => [3,1]
 => [[1,3,4],[2]]
 => 2
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 3
[2,3,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[2,3,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[2,4,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
[3,1,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 3
[3,2,1,4] => [3,1]
 => [[1,3,4],[2]]
 => 2
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
[3,4,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[3,4,2,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
[4,1,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => 2
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => 2
[4,2,3,1] => [3,1]
 => [[1,3,4],[2]]
 => 2
[4,3,1,2] => [3,1]
 => [[1,3,4],[2]]
 => 2
[4,3,2,1] => [4]
 => [[1,2,3,4]]
 => 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,4,5,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,5,3,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,2,5,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
[1,3,4,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,3,4,5,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,3,5,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,2,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
[1,4,2,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
[1,4,3,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
[1,4,5,2,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000157
(load all 6 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion 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] => [1,3,2] => [[1,2],[3]]
 => 1 = 2 - 1
[3,1,2] => [3,1,2] => [[1,2],[3]]
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 2 = 3 - 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] => [1,2,4,3] => [[1,2,3],[4]]
 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
 => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 3 - 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] => [1,3,2,4] => [[1,2,4],[3]]
 => 1 = 2 - 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
 => 1 = 2 - 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2 = 3 - 1
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
 => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
 => 2 = 3 - 1
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
 => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 3 = 4 - 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] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 2 = 3 - 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] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 1 = 2 - 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 2 = 3 - 1
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => 1 = 2 - 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 2 = 3 - 1
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 2 = 3 - 1
[1,4,5,2,3] => [1,3,5,2,4] => [[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: St000211
(load all 15 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ 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}}
 => 1 = 2 - 1
[2,1] => [[1],[2]]
 => {{1},{2}}
 => 0 = 1 - 1
[1,2,3] => [[1,2,3]]
 => {{1,2,3}}
 => 2 = 3 - 1
[1,3,2] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1 = 2 - 1
[2,1,3] => [[1,3],[2]]
 => {{1,3},{2}}
 => 1 = 2 - 1
[2,3,1] => [[1,2],[3]]
 => {{1,2},{3}}
 => 1 = 2 - 1
[3,1,2] => [[1,3],[2]]
 => {{1,3},{2}}
 => 1 = 2 - 1
[3,2,1] => [[1],[2],[3]]
 => {{1},{2},{3}}
 => 0 = 1 - 1
[1,2,3,4] => [[1,2,3,4]]
 => {{1,2,3,4}}
 => 3 = 4 - 1
[1,2,4,3] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[1,3,2,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2 = 3 - 1
[1,3,4,2] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[1,4,2,3] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2 = 3 - 1
[1,4,3,2] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[2,1,3,4] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[2,1,4,3] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2 = 3 - 1
[2,3,1,4] => [[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 2 = 3 - 1
[2,3,4,1] => [[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 2 = 3 - 1
[2,4,1,3] => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2 = 3 - 1
[2,4,3,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[3,1,2,4] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[3,1,4,2] => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2 = 3 - 1
[3,2,1,4] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[3,2,4,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[3,4,1,2] => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2 = 3 - 1
[3,4,2,1] => [[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 1 = 2 - 1
[4,1,2,3] => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2 = 3 - 1
[4,1,3,2] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[4,2,1,3] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[4,2,3,1] => [[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 1 = 2 - 1
[4,3,1,2] => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 1 = 2 - 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 0 = 1 - 1
[1,2,3,4,5] => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 4 = 5 - 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3 = 4 - 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3 = 4 - 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3 = 4 - 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3 = 4 - 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3 = 4 - 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 3 = 4 - 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 3 = 4 - 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 3 = 4 - 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3 = 4 - 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 2 = 3 - 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3 = 4 - 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 3 = 4 - 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 2 = 3 - 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 2 = 3 - 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3 = 4 - 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: St000228
(load all 6 compositions to match this statistic)
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => []
 => 0 = 1 - 1
[1,2] => [1,1]
 => [1]
 => 1 = 2 - 1
[2,1] => [2]
 => []
 => 0 = 1 - 1
[1,2,3] => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,3,2] => [2,1]
 => [1]
 => 1 = 2 - 1
[2,1,3] => [2,1]
 => [1]
 => 1 = 2 - 1
[2,3,1] => [2,1]
 => [1]
 => 1 = 2 - 1
[3,1,2] => [2,1]
 => [1]
 => 1 = 2 - 1
[3,2,1] => [3]
 => []
 => 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,3,4,2] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,4,2,3] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,4,3,2] => [3,1]
 => [1]
 => 1 = 2 - 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[2,1,4,3] => [2,2]
 => [2]
 => 2 = 3 - 1
[2,3,1,4] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[2,3,4,1] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[2,4,1,3] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[2,4,3,1] => [3,1]
 => [1]
 => 1 = 2 - 1
[3,1,2,4] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[3,1,4,2] => [2,2]
 => [2]
 => 2 = 3 - 1
[3,2,1,4] => [3,1]
 => [1]
 => 1 = 2 - 1
[3,2,4,1] => [3,1]
 => [1]
 => 1 = 2 - 1
[3,4,1,2] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[3,4,2,1] => [3,1]
 => [1]
 => 1 = 2 - 1
[4,1,2,3] => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
[4,1,3,2] => [3,1]
 => [1]
 => 1 = 2 - 1
[4,2,1,3] => [3,1]
 => [1]
 => 1 = 2 - 1
[4,2,3,1] => [3,1]
 => [1]
 => 1 = 2 - 1
[4,3,1,2] => [3,1]
 => [1]
 => 1 = 2 - 1
[4,3,2,1] => [4]
 => []
 => 0 = 1 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 4 = 5 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,2,4,5,3] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
[1,3,4,2,5] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,3,4,5,2] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,4,2,3,5] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => 3 = 4 - 1
[1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
[1,4,5,2,3] => [2,1,1,1]
 => [1,1,1]
 => 3 = 4 - 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000377
(load all 2 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => 0 = 1 - 1
[1,2] => [2]
 => [1,1]
 => 1 = 2 - 1
[2,1] => [1,1]
 => [2]
 => 0 = 1 - 1
[1,2,3] => [3]
 => [1,1,1]
 => 2 = 3 - 1
[1,3,2] => [2,1]
 => [3]
 => 1 = 2 - 1
[2,1,3] => [2,1]
 => [3]
 => 1 = 2 - 1
[2,3,1] => [2,1]
 => [3]
 => 1 = 2 - 1
[3,1,2] => [2,1]
 => [3]
 => 1 = 2 - 1
[3,2,1] => [1,1,1]
 => [2,1]
 => 0 = 1 - 1
[1,2,3,4] => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
[1,2,4,3] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[1,3,2,4] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[1,4,3,2] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[2,1,3,4] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[2,1,4,3] => [2,2]
 => [4]
 => 2 = 3 - 1
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[2,3,4,1] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[2,4,1,3] => [2,2]
 => [4]
 => 2 = 3 - 1
[2,4,3,1] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[3,1,4,2] => [2,2]
 => [4]
 => 2 = 3 - 1
[3,2,1,4] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[3,2,4,1] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[3,4,1,2] => [2,2]
 => [4]
 => 2 = 3 - 1
[3,4,2,1] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[4,1,2,3] => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
[4,1,3,2] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[4,2,1,3] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[4,2,3,1] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[4,3,1,2] => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1]
 => [3,1]
 => 0 = 1 - 1
[1,2,3,4,5] => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
[1,2,3,5,4] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,2,4,3,5] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,2,4,5,3] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,2,5,3,4] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
[1,3,2,4,5] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,3,2,5,4] => [3,2]
 => [5]
 => 3 = 4 - 1
[1,3,4,2,5] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,3,5,2,4] => [3,2]
 => [5]
 => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
[1,4,2,3,5] => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
[1,4,2,5,3] => [3,2]
 => [5]
 => 3 = 4 - 1
[1,4,3,2,5] => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
[1,4,3,5,2] => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
[1,4,5,2,3] => [3,2]
 => [5]
 => 3 = 4 - 1
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000010
Mapping
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ 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] => [[1,2]]
 => [1,2] => [1,1]
 => 2
[2,1] => [[1],[2]]
 => [2,1] => [2]
 => 1
[1,2,3] => [[1,2,3]]
 => [1,2,3] => [1,1,1]
 => 3
[1,3,2] => [[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2
[2,1,3] => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2
[2,3,1] => [[1,3],[2]]
 => [2,1,3] => [2,1]
 => 2
[3,1,2] => [[1,2],[3]]
 => [3,1,2] => [2,1]
 => 2
[3,2,1] => [[1],[2],[3]]
 => [3,2,1] => [3]
 => 1
[1,2,3,4] => [[1,2,3,4]]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,2,4,3] => [[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3
[1,3,2,4] => [[1,2,4],[3]]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,3,4,2] => [[1,2,4],[3]]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,4,2,3] => [[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3
[1,4,3,2] => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 2
[2,1,3,4] => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3
[2,1,4,3] => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,1]
 => 3
[2,3,1,4] => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3
[2,3,4,1] => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,1]
 => 3
[2,4,1,3] => [[1,3],[2,4]]
 => [2,4,1,3] => [2,1,1]
 => 3
[2,4,3,1] => [[1,3],[2],[4]]
 => [4,2,1,3] => [3,1]
 => 2
[3,1,2,4] => [[1,2,4],[3]]
 => [3,1,2,4] => [2,1,1]
 => 3
[3,1,4,2] => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3
[3,2,1,4] => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 2
[3,2,4,1] => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 2
[3,4,1,2] => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,1]
 => 3
[3,4,2,1] => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,1]
 => 2
[4,1,2,3] => [[1,2,3],[4]]
 => [4,1,2,3] => [2,1,1]
 => 3
[4,1,3,2] => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 2
[4,2,1,3] => [[1,3],[2],[4]]
 => [4,2,1,3] => [3,1]
 => 2
[4,2,3,1] => [[1,3],[2],[4]]
 => [4,2,1,3] => [3,1]
 => 2
[4,3,1,2] => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,1]
 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4]
 => 1
[1,2,3,4,5] => [[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]]
 => [5,1,2,3,4] => [2,1,1,1]
 => 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,1,1,1]
 => 4
[1,2,4,5,3] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,1,1,1]
 => 4
[1,2,5,3,4] => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,1,1,1]
 => 4
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,1,1]
 => 3
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,1,1,1]
 => 4
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,1,1,1]
 => 4
[1,3,4,2,5] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,1,1,1]
 => 4
[1,3,4,5,2] => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,1,1,1]
 => 4
[1,3,5,2,4] => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,1,1,1]
 => 4
[1,3,5,4,2] => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,1,1]
 => 3
[1,4,2,3,5] => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,1,1,1]
 => 4
[1,4,2,5,3] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,1,1]
 => 4
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,1,1]
 => 3
[1,4,3,5,2] => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,1,1]
 => 3
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,1,1]
 => 4
Description
The length of the partition.
Matching statistic: St000054
(load all 3 compositions to match this statistic)
Mapping
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,2] => [2]
 => [[1,2]]
 => [1,2] => 1
[2,1] => [1,1]
 => [[1],[2]]
 => [2,1] => 2
[1,2,3] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2
[2,3,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2
[3,1,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2
[3,2,1] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[1,2,3,4] => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[1,2,4,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[1,3,2,4] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[2,1,3,4] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 3
[2,3,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[2,3,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[2,4,1,3] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 3
[2,4,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[3,1,2,4] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 3
[3,2,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[3,2,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 3
[3,4,2,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[4,1,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[4,1,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[4,2,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[4,2,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[4,3,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 3
[4,3,2,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 4
[1,2,3,4,5] => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[1,2,3,5,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,2,4,3,5] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,2,4,5,3] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,2,5,3,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,2,5,4,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[1,3,2,4,5] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,3,2,5,4] => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
[1,3,4,2,5] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,3,4,5,2] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,3,5,2,4] => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[1,4,2,3,5] => [4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[1,4,2,5,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
[1,4,3,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[1,4,5,2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000093
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ 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] => ([],2)
 => [2] => ([],2)
 => 2
[2,1] => ([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => [3] => ([],3)
 => 3
[1,3,2] => ([(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => [4] => ([],4)
 => 4
[1,2,4,3] => ([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => ([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => ([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,3,4,5] => ([],5)
 => [5] => ([],5)
 => 5
[1,2,3,5,4] => ([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,3,5] => ([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 4
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 $G$.
The following 83 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000074The number of special entries. St000141The maximum drop size of a permutation. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St000288The number of ones in a binary word. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001726The number of visible inversions of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St000369The dinv deficit of a Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000018The number of inversions of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001298The number of repeated entries in the Lehmer code of a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000703The number of deficiencies of a permutation. St000470The number of runs in a permutation. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000209Maximum difference of elements in cycles. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. 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. St000051The size of the left subtree of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St000956The maximal displacement of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001668The number of points of the poset minus the width of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001060The distinguishing index of a graph. 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. St001330The hat guessing number of a graph. St001896The number of right descents of a signed permutations. St001720The minimal length of a chain of small intervals in a lattice. St001863The number of weak excedances of a signed permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.