Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St001935
Mapping
St001935: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 0
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 0
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 0
[1,1,1,3] => 1
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 0
[1,1,1,4] => 1
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 0
[1,1,2,2] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,2,1,1] => 0
[1,1,2,3] => 2
[1,1,3,2] => 1
[1,2,1,3] => 2
[1,2,3,1] => 2
[1,3,1,2] => 2
[1,3,2,1] => 1
[2,1,1,3] => 1
[2,1,3,1] => 1
[2,3,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 1
Description
The number of ascents in a parking function. This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St000021
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,1] => [1,2] => [1,2] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,1,1] => [1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [2,1,1] => [2,3,1] => [3,1,2] => 1
[1,2,1] => [1,2,1] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [1,1,2] => [1,2,3] => [1,2,3] => 0
[1,1,3] => [3,1,1] => [2,3,1] => [3,1,2] => 1
[1,3,1] => [1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [1,1,3] => [1,2,3] => [1,2,3] => 0
[1,2,2] => [2,2,1] => [3,1,2] => [2,3,1] => 1
[2,1,2] => [2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [1,2,2] => [1,2,3] => [1,2,3] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 2
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 2
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 2
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 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: St000157
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
 => 0
[1,1] => [1,1] => [1,2] => [[1,2]]
 => 0
[1,2] => [2,1] => [2,1] => [[1],[2]]
 => 1
[2,1] => [1,2] => [1,2] => [[1,2]]
 => 0
[1,1,1] => [1,1,1] => [1,2,3] => [[1,2,3]]
 => 0
[1,1,2] => [2,1,1] => [2,3,1] => [[1,3],[2]]
 => 1
[1,2,1] => [1,2,1] => [1,3,2] => [[1,2],[3]]
 => 1
[2,1,1] => [1,1,2] => [1,2,3] => [[1,2,3]]
 => 0
[1,1,3] => [3,1,1] => [2,3,1] => [[1,3],[2]]
 => 1
[1,3,1] => [1,3,1] => [1,3,2] => [[1,2],[3]]
 => 1
[3,1,1] => [1,1,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,2,2] => [2,2,1] => [3,1,2] => [[1,2],[3]]
 => 1
[2,1,2] => [2,1,2] => [2,1,3] => [[1,3],[2]]
 => 1
[2,2,1] => [1,2,2] => [1,2,3] => [[1,2,3]]
 => 0
[1,2,3] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 2
[1,3,2] => [2,3,1] => [3,1,2] => [[1,2],[3]]
 => 1
[2,1,3] => [3,1,2] => [2,3,1] => [[1,3],[2]]
 => 1
[2,3,1] => [1,3,2] => [1,3,2] => [[1,2],[3]]
 => 1
[3,1,2] => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[3,2,1] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
 => 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => [[1,2,3],[4]]
 => 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => [[1,3,4],[2]]
 => 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => [[1,4],[2],[3]]
 => 2
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => [[1,3],[2],[4]]
 => 2
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => [[1,3],[2,4]]
 => 2
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
 => 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => [[1,3,4],[2]]
 => 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => [[1,2,4],[3]]
 => 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => [[1,3,4],[2]]
 => 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000325
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,1] => [1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [2,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,2,1] => [1,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,3] => [3,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,3,1] => [1,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,2] => [2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,1,2] => [2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => [4,3,1,2] => 3 = 2 + 1
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 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
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,1] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,1] => [1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [2,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,2,1] => [1,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,3] => [3,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,3,1] => [1,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,2] => [2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,1,2] => [2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => [4,3,1,2] => 3 = 2 + 1
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 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: St000354
(load all 4 compositions to match this statistic)
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ? = 0
[1,1] => [1,1] => [1,2] => 0
[1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[1,1,1] => [1,1,1] => [1,2,3] => 0
[1,1,2] => [2,1,1] => [2,3,1] => 1
[1,2,1] => [1,2,1] => [1,3,2] => 1
[2,1,1] => [1,1,2] => [1,2,3] => 0
[1,1,3] => [3,1,1] => [2,3,1] => 1
[1,3,1] => [1,3,1] => [1,3,2] => 1
[3,1,1] => [1,1,3] => [1,2,3] => 0
[1,2,2] => [2,2,1] => [3,1,2] => 1
[2,1,2] => [2,1,2] => [2,1,3] => 1
[2,2,1] => [1,2,2] => [1,2,3] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [3,1,2] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => 0
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => 0
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => 0
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => 0
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => 2
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => 0
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => 2
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => 2
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => 2
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => 2
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => 1
[3,2,1,1] => [1,1,2,3] => [1,2,3,4] => 0
Description
The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St001896
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,1] => [1,2] => [1,2] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,1,1] => [1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [2,1,1] => [2,3,1] => [2,3,1] => 1
[1,2,1] => [1,2,1] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [1,1,2] => [1,2,3] => [1,2,3] => 0
[1,1,3] => [3,1,1] => [2,3,1] => [2,3,1] => 1
[1,3,1] => [1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [1,1,3] => [1,2,3] => [1,2,3] => 0
[1,2,2] => [2,2,1] => [3,1,2] => [3,1,2] => 1
[2,1,2] => [2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [1,2,2] => [1,2,3] => [1,2,3] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [2,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,2,1] => [1,2,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[1,2,1,1] => [1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,1,1] => [1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,3] => [3,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,3,1] => [1,3,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,1,1] => [1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,1,1] => [1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,4] => [4,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,4,1] => [1,4,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[1,4,1,1] => [1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,1,1] => [1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,2] => [2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,2,1,2] => [2,1,2,1] => [2,4,1,3] => [2,4,1,3] => 2
[1,2,2,1] => [1,2,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,1,2] => [2,1,1,2] => [2,3,1,4] => [2,3,1,4] => 1
[2,1,2,1] => [1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[2,2,1,1] => [1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,3] => [3,2,1,1] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,3,2] => [2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,2,1,3] => [3,1,2,1] => [2,4,3,1] => [2,4,3,1] => 2
[1,2,3,1] => [1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[1,3,1,2] => [2,1,3,1] => [2,4,1,3] => [2,4,1,3] => 2
[1,3,2,1] => [1,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,1,3] => [3,1,1,2] => [2,3,4,1] => [2,3,4,1] => 1
[2,1,3,1] => [1,3,1,2] => [1,3,4,2] => [1,3,4,2] => 1
[2,3,1,1] => [1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,1,2] => [2,1,1,3] => [2,3,1,4] => [2,3,1,4] => 1
[3,1,2,1] => [1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,1,2] => [2,1,1,1,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,1,1,3] => [3,1,1,1,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,1,1,4] => [4,1,1,1,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,1,1,5] => [5,1,1,1,1] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[1,1,1,2,2] => [2,2,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,2,1,2] => [2,1,2,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,2,1,1,2] => [2,1,1,2,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,1,1,1,2] => [2,1,1,1,2] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,2,3] => [3,2,1,1,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,3,2] => [2,3,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,2,1,3] => [3,1,2,1,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,3,1,2] => [2,1,3,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,2,1,1,3] => [3,1,1,2,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,3,1,1,2] => [2,1,1,3,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,1,1,1,3] => [3,1,1,1,2] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[3,1,1,1,2] => [2,1,1,1,3] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,2,4] => [4,2,1,1,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,4,2] => [2,4,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,2,1,4] => [4,1,2,1,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,4,1,2] => [2,1,4,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,2,1,1,4] => [4,1,1,2,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,4,1,1,2] => [2,1,1,4,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,1,1,1,4] => [4,1,1,1,2] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[4,1,1,1,2] => [2,1,1,1,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,2,5] => [5,2,1,1,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,5,2] => [2,5,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,2,1,5] => [5,1,2,1,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,5,1,2] => [2,1,5,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,2,1,1,5] => [5,1,1,2,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,5,1,1,2] => [2,1,1,5,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[2,1,1,1,5] => [5,1,1,1,2] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[5,1,1,1,2] => [2,1,1,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,3,3] => [3,3,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,3,1,3] => [3,1,3,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,3,1,1,3] => [3,1,1,3,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[3,1,1,1,3] => [3,1,1,1,3] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,3,4] => [4,3,1,1,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,4,3] => [3,4,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,3,1,4] => [4,1,3,1,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,4,1,3] => [3,1,4,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,3,1,1,4] => [4,1,1,3,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,4,1,1,3] => [3,1,1,4,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
[3,1,1,1,4] => [4,1,1,1,3] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 1
[4,1,1,1,3] => [3,1,1,1,4] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
[1,1,1,3,5] => [5,3,1,1,1] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
[1,1,1,5,3] => [3,5,1,1,1] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1
[1,1,3,1,5] => [5,1,3,1,1] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 2
[1,1,5,1,3] => [3,1,5,1,1] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
[1,3,1,1,5] => [5,1,1,3,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 2
[1,5,1,1,3] => [3,1,1,5,1] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
Description
The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St000761
Mapping
Mp00319: Parking functions —to composition⟶ Integer compositions
St000761: Integer compositions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,1,1] => [1,1,1] => 0
[1,1,2] => [1,1,2] => 1
[1,2,1] => [1,2,1] => 1
[2,1,1] => [2,1,1] => 0
[1,1,3] => [1,1,3] => 1
[1,3,1] => [1,3,1] => 1
[3,1,1] => [3,1,1] => 0
[1,2,2] => [1,2,2] => 1
[2,1,2] => [2,1,2] => 1
[2,2,1] => [2,2,1] => 0
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,1,1,1] => 0
[1,1,1,2] => [1,1,1,2] => 1
[1,1,2,1] => [1,1,2,1] => 1
[1,2,1,1] => [1,2,1,1] => 1
[2,1,1,1] => [2,1,1,1] => 0
[1,1,1,3] => [1,1,1,3] => 1
[1,1,3,1] => [1,1,3,1] => 1
[1,3,1,1] => [1,3,1,1] => 1
[3,1,1,1] => [3,1,1,1] => 0
[1,1,1,4] => [1,1,1,4] => 1
[1,1,4,1] => [1,1,4,1] => 1
[1,4,1,1] => [1,4,1,1] => 1
[4,1,1,1] => [4,1,1,1] => 0
[1,1,2,2] => [1,1,2,2] => 1
[1,2,1,2] => [1,2,1,2] => 2
[1,2,2,1] => [1,2,2,1] => 1
[2,1,1,2] => [2,1,1,2] => 1
[2,1,2,1] => [2,1,2,1] => 1
[2,2,1,1] => [2,2,1,1] => 0
[1,1,2,3] => [1,1,2,3] => 2
[1,1,3,2] => [1,1,3,2] => 1
[1,2,1,3] => [1,2,1,3] => 2
[1,2,3,1] => [1,2,3,1] => 2
[1,3,1,2] => [1,3,1,2] => 2
[1,3,2,1] => [1,3,2,1] => 1
[2,1,1,3] => [2,1,1,3] => 1
[2,1,3,1] => [2,1,3,1] => 1
[2,3,1,1] => [2,3,1,1] => 1
[3,1,1,2] => [3,1,1,2] => 1
[3,1,2,1] => [3,1,2,1] => 1
[1,2,3,4] => [1,2,3,4] => ? = 3
[1,2,4,3] => [1,2,4,3] => ? = 2
[1,3,2,4] => [1,3,2,4] => ? = 2
[1,3,4,2] => [1,3,4,2] => ? = 2
[1,4,2,3] => [1,4,2,3] => ? = 2
[1,4,3,2] => [1,4,3,2] => ? = 1
[2,1,3,4] => [2,1,3,4] => ? = 2
[2,1,4,3] => [2,1,4,3] => ? = 1
[2,3,1,4] => [2,3,1,4] => ? = 2
[2,3,4,1] => [2,3,4,1] => ? = 2
[2,4,1,3] => [2,4,1,3] => ? = 2
[2,4,3,1] => [2,4,3,1] => ? = 1
[3,1,2,4] => [3,1,2,4] => ? = 2
[3,1,4,2] => [3,1,4,2] => ? = 1
[3,2,1,4] => [3,2,1,4] => ? = 1
[3,2,4,1] => [3,2,4,1] => ? = 1
[3,4,1,2] => [3,4,1,2] => ? = 2
[3,4,2,1] => [3,4,2,1] => ? = 1
[4,1,2,3] => [4,1,2,3] => ? = 2
[4,1,3,2] => [4,1,3,2] => ? = 1
[4,2,1,3] => [4,2,1,3] => ? = 1
[4,2,3,1] => [4,2,3,1] => ? = 1
[4,3,1,2] => [4,3,1,2] => ? = 1
[4,3,2,1] => [4,3,2,1] => ? = 0
[1,1,1,2,5] => [1,1,1,2,5] => ? = 2
[1,1,1,5,2] => [1,1,1,5,2] => ? = 1
[1,1,2,1,5] => [1,1,2,1,5] => ? = 2
[1,1,2,5,1] => [1,1,2,5,1] => ? = 2
[1,1,5,1,2] => [1,1,5,1,2] => ? = 2
[1,1,5,2,1] => [1,1,5,2,1] => ? = 1
[1,2,1,1,5] => [1,2,1,1,5] => ? = 2
[1,2,1,5,1] => [1,2,1,5,1] => ? = 2
[1,2,5,1,1] => [1,2,5,1,1] => ? = 2
[1,5,1,1,2] => [1,5,1,1,2] => ? = 2
[1,5,1,2,1] => [1,5,1,2,1] => ? = 2
[1,5,2,1,1] => [1,5,2,1,1] => ? = 1
[2,1,1,1,5] => [2,1,1,1,5] => ? = 1
[2,1,1,5,1] => [2,1,1,5,1] => ? = 1
[2,1,5,1,1] => [2,1,5,1,1] => ? = 1
[2,5,1,1,1] => [2,5,1,1,1] => ? = 1
[5,1,1,1,2] => [5,1,1,1,2] => ? = 1
[5,1,1,2,1] => [5,1,1,2,1] => ? = 1
[5,1,2,1,1] => [5,1,2,1,1] => ? = 1
[5,2,1,1,1] => [5,2,1,1,1] => ? = 0
[1,1,1,3,4] => [1,1,1,3,4] => ? = 2
[1,1,1,4,3] => [1,1,1,4,3] => ? = 1
[1,1,3,1,4] => [1,1,3,1,4] => ? = 2
[1,1,3,4,1] => [1,1,3,4,1] => ? = 2
[1,1,4,1,3] => [1,1,4,1,3] => ? = 2
[1,1,4,3,1] => [1,1,4,3,1] => ? = 1
Description
The number of ascents in an integer composition. A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St001946
Mapping
Mp00320: Parking functions —reverse⟶ Parking functions
St001946: Parking functions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[1,2] => [2,1] => 1
[2,1] => [1,2] => 0
[1,1,1] => [1,1,1] => 0
[1,1,2] => [2,1,1] => 1
[1,2,1] => [1,2,1] => 1
[2,1,1] => [1,1,2] => 0
[1,1,3] => [3,1,1] => 1
[1,3,1] => [1,3,1] => 1
[3,1,1] => [1,1,3] => 0
[1,2,2] => [2,2,1] => 1
[2,1,2] => [2,1,2] => 1
[2,2,1] => [1,2,2] => 0
[1,2,3] => [3,2,1] => 2
[1,3,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 0
[1,1,1,1] => [1,1,1,1] => 0
[1,1,1,2] => [2,1,1,1] => 1
[1,1,2,1] => [1,2,1,1] => 1
[1,2,1,1] => [1,1,2,1] => 1
[2,1,1,1] => [1,1,1,2] => 0
[1,1,1,3] => [3,1,1,1] => 1
[1,1,3,1] => [1,3,1,1] => 1
[1,3,1,1] => [1,1,3,1] => 1
[3,1,1,1] => [1,1,1,3] => 0
[1,1,1,4] => [4,1,1,1] => 1
[1,1,4,1] => [1,4,1,1] => 1
[1,4,1,1] => [1,1,4,1] => 1
[4,1,1,1] => [1,1,1,4] => 0
[1,1,2,2] => [2,2,1,1] => 1
[1,2,1,2] => [2,1,2,1] => 2
[1,2,2,1] => [1,2,2,1] => 1
[2,1,1,2] => [2,1,1,2] => 1
[2,1,2,1] => [1,2,1,2] => 1
[2,2,1,1] => [1,1,2,2] => 0
[1,1,2,3] => [3,2,1,1] => 2
[1,1,3,2] => [2,3,1,1] => 1
[1,2,1,3] => [3,1,2,1] => 2
[1,2,3,1] => [1,3,2,1] => 2
[1,3,1,2] => [2,1,3,1] => 2
[1,3,2,1] => [1,2,3,1] => 1
[2,1,1,3] => [3,1,1,2] => 1
[2,1,3,1] => [1,3,1,2] => 1
[2,3,1,1] => [1,1,3,2] => 1
[3,1,1,2] => [2,1,1,3] => 1
[3,1,2,1] => [1,2,1,3] => 1
[1,1,1,1,1] => [1,1,1,1,1] => ? = 0
[1,1,1,1,2] => [2,1,1,1,1] => ? = 1
[1,1,1,2,1] => [1,2,1,1,1] => ? = 1
[1,1,2,1,1] => [1,1,2,1,1] => ? = 1
[1,2,1,1,1] => [1,1,1,2,1] => ? = 1
[2,1,1,1,1] => [1,1,1,1,2] => ? = 0
[1,1,1,1,3] => [3,1,1,1,1] => ? = 1
[1,1,1,3,1] => [1,3,1,1,1] => ? = 1
[1,1,3,1,1] => [1,1,3,1,1] => ? = 1
[1,3,1,1,1] => [1,1,1,3,1] => ? = 1
[3,1,1,1,1] => [1,1,1,1,3] => ? = 0
[1,1,1,1,4] => [4,1,1,1,1] => ? = 1
[1,1,1,4,1] => [1,4,1,1,1] => ? = 1
[1,1,4,1,1] => [1,1,4,1,1] => ? = 1
[1,4,1,1,1] => [1,1,1,4,1] => ? = 1
[4,1,1,1,1] => [1,1,1,1,4] => ? = 0
[1,1,1,1,5] => [5,1,1,1,1] => ? = 1
[1,1,1,5,1] => [1,5,1,1,1] => ? = 1
[1,1,5,1,1] => [1,1,5,1,1] => ? = 1
[1,5,1,1,1] => [1,1,1,5,1] => ? = 1
[5,1,1,1,1] => [1,1,1,1,5] => ? = 0
[1,1,1,2,2] => [2,2,1,1,1] => ? = 1
[1,1,2,1,2] => [2,1,2,1,1] => ? = 2
[1,1,2,2,1] => [1,2,2,1,1] => ? = 1
[1,2,1,1,2] => [2,1,1,2,1] => ? = 2
[1,2,1,2,1] => [1,2,1,2,1] => ? = 2
[1,2,2,1,1] => [1,1,2,2,1] => ? = 1
[2,1,1,1,2] => [2,1,1,1,2] => ? = 1
[2,1,1,2,1] => [1,2,1,1,2] => ? = 1
[2,1,2,1,1] => [1,1,2,1,2] => ? = 1
[2,2,1,1,1] => [1,1,1,2,2] => ? = 0
[1,1,1,2,3] => [3,2,1,1,1] => ? = 2
[1,1,1,3,2] => [2,3,1,1,1] => ? = 1
[1,1,2,1,3] => [3,1,2,1,1] => ? = 2
[1,1,2,3,1] => [1,3,2,1,1] => ? = 2
[1,1,3,1,2] => [2,1,3,1,1] => ? = 2
[1,1,3,2,1] => [1,2,3,1,1] => ? = 1
[1,2,1,1,3] => [3,1,1,2,1] => ? = 2
[1,2,1,3,1] => [1,3,1,2,1] => ? = 2
[1,2,3,1,1] => [1,1,3,2,1] => ? = 2
[1,3,1,1,2] => [2,1,1,3,1] => ? = 2
[1,3,1,2,1] => [1,2,1,3,1] => ? = 2
[1,3,2,1,1] => [1,1,2,3,1] => ? = 1
[2,1,1,1,3] => [3,1,1,1,2] => ? = 1
[2,1,1,3,1] => [1,3,1,1,2] => ? = 1
[2,1,3,1,1] => [1,1,3,1,2] => ? = 1
[2,3,1,1,1] => [1,1,1,3,2] => ? = 1
[3,1,1,1,2] => [2,1,1,1,3] => ? = 1
[3,1,1,2,1] => [1,2,1,1,3] => ? = 1
[3,1,2,1,1] => [1,1,2,1,3] => ? = 1
Description
The number of descents in a parking function. This is the number of indices $i$ such that $p_i > p_{i+1}$.