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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000213
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 4
[1,4,2,3,5] => 3
[1,4,2,5,3] => 3
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 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 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => 1 = 2 - 1
[2,1] => [2,1] => 0 = 1 - 1
[1,2,3] => [1,2,3] => 2 = 3 - 1
[1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [3,2,1] => 0 = 1 - 1
[3,2,1] => [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => 1 = 2 - 1
[1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[2,4,1,3] => [4,3,1,2] => 1 = 2 - 1
[2,4,3,1] => [3,4,1,2] => 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => 1 = 2 - 1
[3,1,4,2] => [4,2,1,3] => 1 = 2 - 1
[3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[3,2,4,1] => [2,4,1,3] => 2 = 3 - 1
[3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[4,1,2,3] => [4,3,2,1] => 0 = 1 - 1
[4,1,3,2] => [3,4,2,1] => 1 = 2 - 1
[4,2,1,3] => [2,4,3,1] => 1 = 2 - 1
[4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
[4,3,1,2] => [4,2,3,1] => 1 = 2 - 1
[4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,5,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,4,5,3] => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => 3 = 4 - 1
[1,3,5,2,4] => [1,5,4,2,3] => 2 = 3 - 1
[1,3,5,4,2] => [1,4,5,2,3] => 3 = 4 - 1
[1,4,2,3,5] => [1,4,3,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,2,4] => 2 = 3 - 1
[1,4,3,2,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,4,3,5,2] => [1,3,5,2,4] => 3 = 4 - 1
[1,4,5,2,3] => [1,4,2,5,3] => 2 = 3 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000010
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
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%
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
[1,2] => [1,2] => [1,1]
=> 2
[2,1] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [3,1,2] => [2,1]
=> 2
[3,1,2] => [3,2,1] => [3]
=> 1
[3,2,1] => [2,3,1] => [2,1]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2] => [1,4,2,3] => [2,1,1]
=> 3
[1,4,2,3] => [1,4,3,2] => [3,1]
=> 2
[1,4,3,2] => [1,3,4,2] => [2,1,1]
=> 3
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1,2,4] => [2,1,1]
=> 3
[2,3,4,1] => [4,1,2,3] => [2,1,1]
=> 3
[2,4,1,3] => [4,3,1,2] => [3,1]
=> 2
[2,4,3,1] => [3,4,1,2] => [2,1,1]
=> 3
[3,1,2,4] => [3,2,1,4] => [3,1]
=> 2
[3,1,4,2] => [4,2,1,3] => [3,1]
=> 2
[3,2,1,4] => [2,3,1,4] => [2,1,1]
=> 3
[3,2,4,1] => [2,4,1,3] => [2,1,1]
=> 3
[3,4,1,2] => [3,1,4,2] => [2,2]
=> 2
[3,4,2,1] => [4,1,3,2] => [3,1]
=> 2
[4,1,2,3] => [4,3,2,1] => [4]
=> 1
[4,1,3,2] => [3,4,2,1] => [3,1]
=> 2
[4,2,1,3] => [2,4,3,1] => [3,1]
=> 2
[4,2,3,1] => [2,3,4,1] => [2,1,1]
=> 3
[4,3,1,2] => [4,2,3,1] => [3,1]
=> 2
[4,3,2,1] => [3,2,4,1] => [3,1]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
=> 4
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => [2,1,1,1]
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
=> 4
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => [3,1,1]
=> 3
[1,3,5,4,2] => [1,4,5,2,3] => [2,1,1,1]
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => [3,1,1]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [2,1,1,1]
=> 4
[1,4,3,5,2] => [1,3,5,2,4] => [2,1,1,1]
=> 4
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000105
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
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%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> 1
[1,2] => [1,2] => {{1},{2}}
=> 2
[2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2
[3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [3,1,4,2] => {{1,3,4},{2}}
=> 2
[2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => [2,4,1,3] => {{1,2,4},{3}}
=> 2
[3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => [4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[4,1,3,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 2
[4,2,1,3] => [3,2,4,1] => {{1,3,4},{2}}
=> 2
[4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,3,1,2] => [3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 4
[1,2,5,3,4] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 4
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 4
[1,3,5,2,4] => [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 4
[1,4,2,3,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3
[1,4,2,5,3] => [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,4,3,5,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4
[1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000325
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,3,1] => 2
[2,3,1] => [3,1,2] => [1,3,2] => 2
[3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1] => [2,3,1] => [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 3
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 2
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 3
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 3
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 2
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 3
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 2
[3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 2
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 3
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 3
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 2
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 1
[4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 2
[4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 2
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2
[4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 4
[1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 3
[1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 4
[1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 3
[1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 4
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 3
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 4
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 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 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,3,1] => 2
[2,3,1] => [3,1,2] => [1,3,2] => 2
[3,1,2] => [3,2,1] => [1,2,3] => 1
[3,2,1] => [2,3,1] => [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 3
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 2
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 3
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 3
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 3
[2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 2
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 3
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 2
[3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 2
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 3
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 3
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 2
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 1
[4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 2
[4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 2
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 2
[4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 4
[1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 3
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 3
[1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 4
[1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 4
[1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 3
[1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 4
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 3
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 4
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 4
[1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 3
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000507
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
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%
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] => [1] => [[1]]
=> 1
[1,2] => [1,2] => [[1,2]]
=> 2
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,3,1] => [3,1,2] => [[1,3],[2]]
=> 2
[3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[3,2,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,3,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 3
[1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 3
[2,3,4,1] => [4,1,2,3] => [[1,3,4],[2]]
=> 3
[2,4,1,3] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[2,4,3,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[3,2,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 3
[3,2,4,1] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[4,1,2,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[4,1,3,2] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[4,3,1,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,3,2,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 4
[1,2,5,3,4] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 4
[1,3,4,5,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 3
[1,3,5,4,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 4
[1,4,3,5,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 4
[1,4,5,2,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 3
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000021
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000021: Permutations ⟶ 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] => [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,1,2] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 2 = 3 - 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[2,4,1,3] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[2,4,3,1] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[3,1,2,4] => [3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[3,1,4,2] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[3,2,4,1] => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 1 = 2 - 1
[3,4,2,1] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,1,2,3] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[4,2,1,3] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[4,3,1,2] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 4 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [5,1,2,4,3] => 2 = 3 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [5,2,1,4,3] => 3 = 4 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2 = 3 - 1
[1,4,2,5,3] => [1,5,3,2,4] => [5,1,3,4,2] => 2 = 3 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [5,3,1,4,2] => 3 = 4 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 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 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ 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] => [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,3,2] => [1,3,2] => [3,2,1] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1,3] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 2 = 3 - 1
[1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 2 = 3 - 1
[2,3,4,1] => [4,1,2,3] => [3,4,1,2] => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 1 = 2 - 1
[2,4,3,1] => [4,1,3,2] => [4,3,1,2] => 2 = 3 - 1
[3,1,2,4] => [2,3,1,4] => [1,2,4,3] => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [1,4,2,3] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 2 = 3 - 1
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [4,2,1,3] => 1 = 2 - 1
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,3,2] => [2,4,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [2,1,3,4] => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => 2 = 3 - 1
[4,3,1,2] => [3,4,2,1] => [3,1,2,4] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,4,5,3,1] => 3 = 4 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,5,3,4,1] => 2 = 3 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => 3 = 4 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [3,4,2,5,1] => 3 = 4 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [3,4,5,2,1] => 3 = 4 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [3,5,2,4,1] => 2 = 3 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [3,5,4,2,1] => 3 = 4 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2 = 3 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [4,2,5,3,1] => 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 3 = 4 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [4,3,5,2,1] => 3 = 4 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,1] => 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: St000015
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,3,4,1] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,4,1,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,3,1] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,1,2,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,4,1] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,3,1,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,5,3,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,4,5,2] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,3,5,4,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,4,2,3,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,4,3,5,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,4,5,2,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
Description
The number of peaks of a Dyck path.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000031The number of cycles in the cycle decomposition of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000167The number of leaves of an ordered tree. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St001480The number of simple summands of the module J^2/J^3. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001946The number of descents in a parking function.
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