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Matching statistic: St000155
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
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: St000021
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,1,2] => 1
[3,2,1] => [2,3,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,3,4,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,3,2,1] => 3
[2,4,1,3] => [4,2,1,3] => 2
[2,4,3,1] => [3,4,2,1] => 2
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [4,3,1,2] => 2
[3,2,1,4] => [2,3,1,4] => 1
[3,2,4,1] => [2,4,3,1] => 2
[3,4,1,2] => [4,1,3,2] => 2
[3,4,2,1] => [4,2,3,1] => 2
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [3,4,1,2] => 1
[4,2,1,3] => [2,4,1,3] => 1
[4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [3,1,4,2] => 2
[4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,5,3,2,4] => 2
[1,3,5,4,2] => [1,4,5,3,2] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,5,4,2,3] => 2
[1,4,3,2,5] => [1,3,4,2,5] => 1
[1,4,3,5,2] => [1,3,5,4,2] => 2
[1,4,5,2,3] => [1,5,2,4,3] => 2
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: St000211
(load all 168 compositions to match this statistic)
(load all 168 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 0
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => {{1,3},{2}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => {{1,2,4},{3}}
=> 2
[2,4,3,1] => {{1,2,4},{3}}
=> 2
[3,1,2,4] => {{1,3},{2},{4}}
=> 1
[3,1,4,2] => {{1,3,4},{2}}
=> 2
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> 1
[4,1,3,2] => {{1,4},{2},{3}}
=> 1
[4,2,1,3] => {{1,4},{2},{3}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 1
[4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 2
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000703
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 2
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => 3
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [4,1,3,2] => 2
[3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,1,3] => 2
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 2
[4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [3,4,2,1] => 2
[4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,3,4] => 2
[1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => 3
[1,3,5,2,4] => [1,4,2,5,3] => 2
[1,3,5,4,2] => [1,5,2,4,3] => 2
[1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,5,2,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,5,3,2,4] => 2
[1,4,5,2,3] => [1,4,5,2,3] => 2
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000325
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000325: Permutations ⟶ 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
[2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,2,1] => 3 = 2 + 1
[3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => 3 = 2 + 1
[2,3,4,1] => [4,3,2,1] => 4 = 3 + 1
[2,4,1,3] => [4,2,1,3] => 3 = 2 + 1
[2,4,3,1] => [3,4,2,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => [4,3,1,2] => 3 = 2 + 1
[3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[3,2,4,1] => [2,4,3,1] => 3 = 2 + 1
[3,4,1,2] => [4,1,3,2] => 3 = 2 + 1
[3,4,2,1] => [4,2,3,1] => 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[4,1,3,2] => [3,4,1,2] => 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[4,3,1,2] => [3,1,4,2] => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,3,4,5,2] => [1,5,4,3,2] => 4 = 3 + 1
[1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,3,2] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,2,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,4,2] => 3 = 2 + 1
[1,4,5,2,3] => [1,5,2,4,3] => 3 = 2 + 1
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 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000470: Permutations ⟶ 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
[2,1] => [2,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,2,1] => 3 = 2 + 1
[3,1,2] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [2,3,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => [3,2,1,4] => 3 = 2 + 1
[2,3,4,1] => [4,3,2,1] => 4 = 3 + 1
[2,4,1,3] => [4,2,1,3] => 3 = 2 + 1
[2,4,3,1] => [3,4,2,1] => 3 = 2 + 1
[3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => [4,3,1,2] => 3 = 2 + 1
[3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[3,2,4,1] => [2,4,3,1] => 3 = 2 + 1
[3,4,1,2] => [4,1,3,2] => 3 = 2 + 1
[3,4,2,1] => [4,2,3,1] => 3 = 2 + 1
[4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[4,1,3,2] => [3,4,1,2] => 2 = 1 + 1
[4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[4,3,1,2] => [3,1,4,2] => 3 = 2 + 1
[4,3,2,1] => [3,2,4,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,3,4,5,2] => [1,5,4,3,2] => 4 = 3 + 1
[1,3,5,2,4] => [1,5,3,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [1,4,5,3,2] => 3 = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,2,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,4,3,5,2] => [1,3,5,4,2] => 3 = 2 + 1
[1,4,5,2,3] => [1,5,2,4,3] => 3 = 2 + 1
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: St000157
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: 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
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[3,1,2] => [3,1,2] => [[1,3],[2]]
=> 1
[3,2,1] => [2,3,1] => [[1,2],[3]]
=> 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,4,3,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[2,3,4,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[2,4,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[2,4,3,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[3,1,4,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[3,2,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[3,2,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,4,1,2] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[3,4,2,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,1,2,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[4,1,3,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[4,2,1,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[4,2,3,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[4,3,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[4,3,2,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 3
[1,3,5,2,4] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 2
[1,3,5,4,2] => [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,4,2,5,3] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 2
[1,4,3,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,4,3,5,2] => [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 2
[1,4,5,2,3] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000245
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,2,1] => [1,2,3] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [2,3,1] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 2
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 1
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,1,4] => [2,3,4,1] => 2
[2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 3
[2,4,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[2,4,3,1] => [3,4,2,1] => [2,1,3,4] => 2
[3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [4,3,1,2] => [1,2,4,3] => 2
[3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 1
[3,2,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => 2
[3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 2
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => 1
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 1
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => 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] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 3
[1,3,5,2,4] => [1,5,3,2,4] => [5,1,3,4,2] => 2
[1,3,5,4,2] => [1,4,5,3,2] => [5,2,1,3,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [5,1,2,4,3] => 2
[1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 1
[1,4,3,5,2] => [1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000662
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [2,3,1] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 2
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => 1
[1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 3
[2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 2
[3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[3,2,4,1] => [2,4,3,1] => [4,2,3,1] => 2
[3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 2
[3,4,2,1] => [4,2,3,1] => [4,1,3,2] => 2
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 1
[4,1,3,2] => [3,4,1,2] => [1,4,2,3] => 1
[4,2,1,3] => [2,4,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 1
[4,3,1,2] => [3,1,4,2] => [3,4,1,2] => 2
[4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 1
[1,2,5,4,3] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [3,4,2,1,5] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [4,5,3,2,1] => 3
[1,3,5,2,4] => [1,5,3,2,4] => [3,4,1,5,2] => 2
[1,3,5,4,2] => [1,4,5,3,2] => [3,5,4,1,2] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => 1
[1,4,2,5,3] => [1,5,4,2,3] => [3,1,5,4,2] => 2
[1,4,3,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => 1
[1,4,3,5,2] => [1,3,5,4,2] => [3,5,2,4,1] => 2
[1,4,5,2,3] => [1,5,2,4,3] => [3,2,5,4,1] => 2
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000105
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
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 = 0 + 1
[1,2] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[2,1] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,2,3] => [2,3,1] => {{1,2,3}}
=> 1 = 0 + 1
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 2 = 1 + 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,4,2,3] => [2,4,1,3] => {{1,2,4},{3}}
=> 2 = 1 + 1
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 2 = 1 + 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[3,1,2,4] => [3,4,2,1] => {{1,3},{2,4}}
=> 2 = 1 + 1
[3,1,4,2] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 2 = 1 + 1
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[3,4,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 2 = 1 + 1
[4,1,3,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 2 = 1 + 1
[4,2,1,3] => [4,3,1,2] => {{1,4},{2,3}}
=> 2 = 1 + 1
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 1 + 1
[4,3,1,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2 = 1 + 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 2 = 1 + 1
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 2 = 1 + 1
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 2 = 1 + 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3 = 2 + 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,4},{3,5}}
=> 2 = 1 + 1
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2 = 1 + 1
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3 = 2 + 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 3 = 2 + 1
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].
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000213The number of weak exceedances (also weak excedences) of a permutation. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000168The number of internal nodes of an ordered tree. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. 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. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001812The biclique partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001935The number of ascents in a parking function.
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