Your data matches 12 different statistics following compositions of up to 3 maps.
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Mp00252: Permutations restrictionPermutations
St000213: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 2
[1,3,2] => [1,2] => 2
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 2
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 3
[1,2,4,3] => [1,2,3] => 3
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 3
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => 3
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => 4
[1,2,3,5,4] => [1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3] => 3
[1,2,4,5,3] => [1,2,4,3] => 3
[1,2,5,3,4] => [1,2,3,4] => 4
[1,2,5,4,3] => [1,2,4,3] => 3
[1,3,2,4,5] => [1,3,2,4] => 3
[1,3,2,5,4] => [1,3,2,4] => 3
[1,3,4,2,5] => [1,3,4,2] => 3
[1,3,4,5,2] => [1,3,4,2] => 3
[1,3,5,2,4] => [1,3,2,4] => 3
[1,3,5,4,2] => [1,3,4,2] => 3
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 3
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
St000245: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 1 = 2 - 1
[1,3,2] => [1,2] => [1,2] => 1 = 2 - 1
[2,1,3] => [2,1] => [2,1] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 1 = 2 - 1
[3,2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[3,2,4,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[3,4,1,2] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [3,2,1] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000010
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00204: Permutations LLPSInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1]
=> 1
[2,1] => [1] => [1] => [1]
=> 1
[1,2,3] => [1,2] => [1,2] => [1,1]
=> 2
[1,3,2] => [1,2] => [1,2] => [1,1]
=> 2
[2,1,3] => [2,1] => [2,1] => [2]
=> 1
[2,3,1] => [2,1] => [2,1] => [2]
=> 1
[3,1,2] => [1,2] => [1,2] => [1,1]
=> 2
[3,2,1] => [2,1] => [2,1] => [2]
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1,4] => [2,3,1] => [3,1,2] => [2,1]
=> 2
[2,3,4,1] => [2,3,1] => [3,1,2] => [2,1]
=> 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,4,3,1] => [2,3,1] => [3,1,2] => [2,1]
=> 2
[3,1,2,4] => [3,1,2] => [3,2,1] => [3]
=> 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [3]
=> 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [2,1]
=> 2
[3,2,4,1] => [3,2,1] => [2,3,1] => [2,1]
=> 2
[3,4,1,2] => [3,1,2] => [3,2,1] => [3]
=> 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [2,1]
=> 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[4,2,3,1] => [2,3,1] => [3,1,2] => [2,1]
=> 2
[4,3,1,2] => [3,1,2] => [3,2,1] => [3]
=> 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [2,1]
=> 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 3
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [3,1]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [3,1]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [3,1]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000105
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
Mp00240: Permutations weak exceedance partitionSet partitions
St000105: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => {{1}}
=> 1
[2,1] => [1] => [1] => {{1}}
=> 1
[1,2,3] => [1,2] => [1,2] => {{1},{2}}
=> 2
[1,3,2] => [1,2] => [1,2] => {{1},{2}}
=> 2
[2,1,3] => [2,1] => [2,1] => {{1,2}}
=> 1
[2,3,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[3,1,2] => [1,2] => [1,2] => {{1},{2}}
=> 2
[3,2,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,2,4,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2,4] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,3,4,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,4,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,4,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[2,1,3,4] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,1,4,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,3,1,4] => [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2
[2,3,4,1] => [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2
[2,4,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,4,3,1] => [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2
[3,1,2,4] => [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,1,4,2] => [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,2,1,4] => [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,4,1] => [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,4,1,2] => [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,4,2,1] => [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[4,1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[4,1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[4,2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[4,2,3,1] => [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2
[4,3,1,2] => [3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[4,3,2,1] => [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 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].
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00064: Permutations reversePermutations
St000325: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 1
[2,1] => [1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 2
[1,3,2] => [1,2] => [1,2] => [2,1] => 2
[2,1,3] => [2,1] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 2
[3,2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,1,4,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1,4] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[2,3,4,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,4,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[3,2,4,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[3,4,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[4,2,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[4,3,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 3
Description
The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00064: Permutations reversePermutations
St000470: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 1
[2,1] => [1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 2
[1,3,2] => [1,2] => [1,2] => [2,1] => 2
[2,1,3] => [2,1] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 2
[3,2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,1,4,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1,4] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[2,3,4,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[2,4,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[3,1,2,4] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[3,2,4,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[3,4,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 3
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 2
[4,2,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 2
[4,3,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 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: St000507
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [[1]]
=> 1
[2,1] => [1] => [1] => [[1]]
=> 1
[1,2,3] => [1,2] => [1,2] => [[1,2]]
=> 2
[1,3,2] => [1,2] => [1,2] => [[1,2]]
=> 2
[2,1,3] => [2,1] => [2,1] => [[1],[2]]
=> 1
[2,3,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[3,1,2] => [1,2] => [1,2] => [[1,2]]
=> 2
[3,2,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,2,4,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,3,2,4] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,4,3,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[2,1,3,4] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,1,4,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,3,1,4] => [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[2,3,4,1] => [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[2,4,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,4,3,1] => [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[3,1,2,4] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[3,2,4,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[3,4,1,2] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[4,1,3,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[4,2,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[4,2,3,1] => [2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[4,3,1,2] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 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).
Mp00252: Permutations restrictionPermutations
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00064: Permutations reversePermutations
St000021: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [2,3,1] => [1,3,2] => 1 = 2 - 1
[3,2,4,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1 = 2 - 1
[3,4,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1 = 2 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [3,1,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,1,2] => [2,1,3] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 3 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 2 = 3 - 1
Description
The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
Mp00088: Permutations Kreweras complementPermutations
St000155: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[2,1,3] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[3,2,1] => [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[2,3,4,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[3,4,1,2] => [3,1,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 3 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => [2,4,1,3] => 2 = 3 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => [2,4,1,3] => 2 = 3 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => [2,4,1,3] => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => [2,1,3,4] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => [2,1,3,4] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 3 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 3 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => [2,1,3,4] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 2 = 3 - 1
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Mp00252: Permutations restrictionPermutations
Mp00066: Permutations inversePermutations
St000702: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => ? = 1
[2,1] => [1] => [1] => ? = 1
[1,2,3] => [1,2] => [1,2] => 2
[1,3,2] => [1,2] => [1,2] => 2
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 2
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 3
[1,2,4,3] => [1,2,3] => [1,2,3] => 3
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 3
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => [3,1,2] => 2
[2,3,4,1] => [2,3,1] => [3,1,2] => 2
[2,4,1,3] => [2,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => [3,1,2] => 2
[3,1,2,4] => [3,1,2] => [2,3,1] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 3
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => [3,1,2] => 2
[4,3,1,2] => [3,1,2] => [2,3,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 3
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 3
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 3
[1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 3
[1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3
[1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[1,5,2,4,3] => [1,2,4,3] => [1,2,4,3] => 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]].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000083The number of left oriented leafs of a binary tree except the first one. St001863The number of weak excedances of a signed permutation.