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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000213
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(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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 $\sigma$.
Matching statistic: St000245
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
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 —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ 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
[1,2] => [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 —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [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].
Matching statistic: St000325
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
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.
Matching statistic: St000470
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
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 —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
Mp00087: Permutations —inverse first fundamental transformation⟶ 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%
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).
Matching statistic: St000021
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
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].
Matching statistic: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
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]].
Matching statistic: St000702
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
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]].
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