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Your data matches 139 different statistics following compositions of up to 3 maps.
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Matching statistic: St001905
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001905: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 1
[1,1,1] => 2
[1,1,2] => 2
[1,2,1] => 1
[2,1,1] => 2
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 2
[1,2,2] => 1
[2,1,2] => 2
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,1,1,1] => 3
[1,1,1,2] => 3
[1,1,2,1] => 3
[1,2,1,1] => 2
[2,1,1,1] => 3
[1,1,1,3] => 3
[1,1,3,1] => 2
[1,3,1,1] => 2
[3,1,1,1] => 3
[1,1,1,4] => 2
[1,1,4,1] => 2
[1,4,1,1] => 2
[4,1,1,1] => 3
[1,1,2,2] => 3
[1,2,1,2] => 2
[1,2,2,1] => 2
[2,1,1,2] => 3
[2,1,2,1] => 3
[2,2,1,1] => 2
[1,1,2,3] => 3
[1,1,3,2] => 2
[1,2,1,3] => 2
[1,2,3,1] => 1
[1,3,1,2] => 2
[1,3,2,1] => 2
[2,1,1,3] => 3
[2,1,3,1] => 2
[2,3,1,1] => 2
[3,1,1,2] => 3
[3,1,2,1] => 3
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.
Matching statistic: St001298
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001298: 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
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => 2
[1,1,2] => [1,2,3] => 2
[1,2,1] => [1,2,3] => 2
[2,1,1] => [2,1,3] => 1
[1,1,3] => [1,2,3] => 2
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 2
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 2
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => 3
[1,1,1,2] => [1,2,3,4] => 3
[1,1,2,1] => [1,2,3,4] => 3
[1,2,1,1] => [1,2,3,4] => 3
[2,1,1,1] => [2,1,3,4] => 2
[1,1,1,3] => [1,2,3,4] => 3
[1,1,3,1] => [1,2,3,4] => 3
[1,3,1,1] => [1,3,2,4] => 2
[3,1,1,1] => [2,3,1,4] => 2
[1,1,1,4] => [1,2,3,4] => 3
[1,1,4,1] => [1,2,4,3] => 2
[1,4,1,1] => [1,3,4,2] => 2
[4,1,1,1] => [2,3,4,1] => 2
[1,1,2,2] => [1,2,3,4] => 3
[1,2,1,2] => [1,2,3,4] => 3
[1,2,2,1] => [1,2,3,4] => 3
[2,1,1,2] => [2,1,3,4] => 2
[2,1,2,1] => [2,1,3,4] => 2
[2,2,1,1] => [3,1,2,4] => 2
[1,1,2,3] => [1,2,3,4] => 3
[1,1,3,2] => [1,2,3,4] => 3
[1,2,1,3] => [1,2,3,4] => 3
[1,2,3,1] => [1,2,3,4] => 3
[1,3,1,2] => [1,3,2,4] => 2
[1,3,2,1] => [1,3,2,4] => 2
[2,1,1,3] => [2,1,3,4] => 2
[2,1,3,1] => [2,1,3,4] => 2
[2,3,1,1] => [3,1,2,4] => 2
[3,1,1,2] => [2,3,1,4] => 2
[3,1,2,1] => [2,3,1,4] => 2
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St000245
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000245: 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] => 1
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => 2
[1,1,2] => [1,2,3] => [1,2,3] => 2
[1,2,1] => [1,2,3] => [1,2,3] => 2
[2,1,1] => [2,1,3] => [2,1,3] => 1
[1,1,3] => [1,2,3] => [1,2,3] => 2
[1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [2,3,1] => [3,1,2] => 1
[1,2,2] => [1,2,3] => [1,2,3] => 2
[2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [3,1,2] => [2,3,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 2
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 3
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 3
[1,1,2,1] => [1,2,3,4] => [1,2,3,4] => 3
[1,2,1,1] => [1,2,3,4] => [1,2,3,4] => 3
[2,1,1,1] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 3
[1,1,3,1] => [1,2,3,4] => [1,2,3,4] => 3
[1,3,1,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,1,1] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 3
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 3
[1,2,1,2] => [1,2,3,4] => [1,2,3,4] => 3
[1,2,2,1] => [1,2,3,4] => [1,2,3,4] => 3
[2,1,1,2] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,2,1] => [2,1,3,4] => [2,1,3,4] => 2
[2,2,1,1] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 3
[1,1,3,2] => [1,2,3,4] => [1,2,3,4] => 3
[1,2,1,3] => [1,2,3,4] => [1,2,3,4] => 3
[1,2,3,1] => [1,2,3,4] => [1,2,3,4] => 3
[1,3,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[2,1,1,3] => [2,1,3,4] => [2,1,3,4] => 2
[2,1,3,1] => [2,1,3,4] => [2,1,3,4] => 2
[2,3,1,1] => [3,1,2,4] => [2,3,1,4] => 2
[3,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2
[3,1,2,1] => [2,3,1,4] => [3,1,2,4] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000662
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000662: 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] => [2,1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,1,1] => [1,2,3] => [3,2,1] => 2
[1,1,2] => [1,2,3] => [3,2,1] => 2
[1,2,1] => [1,2,3] => [3,2,1] => 2
[2,1,1] => [2,1,3] => [2,3,1] => 1
[1,1,3] => [1,2,3] => [3,2,1] => 2
[1,3,1] => [1,3,2] => [3,1,2] => 1
[3,1,1] => [2,3,1] => [2,1,3] => 1
[1,2,2] => [1,2,3] => [3,2,1] => 2
[2,1,2] => [2,1,3] => [2,3,1] => 1
[2,2,1] => [3,1,2] => [1,3,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 2
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 3
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => 3
[2,1,1,1] => [2,1,3,4] => [3,4,2,1] => 2
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 3
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,1,1] => [2,3,1,4] => [3,2,4,1] => 2
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,1,4,1] => [1,2,4,3] => [4,3,1,2] => 2
[1,4,1,1] => [1,3,4,2] => [4,2,1,3] => 2
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => 3
[2,1,1,2] => [2,1,3,4] => [3,4,2,1] => 2
[2,1,2,1] => [2,1,3,4] => [3,4,2,1] => 2
[2,2,1,1] => [3,1,2,4] => [2,4,3,1] => 2
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 3
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => 3
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => 3
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => 2
[2,1,1,3] => [2,1,3,4] => [3,4,2,1] => 2
[2,1,3,1] => [2,1,3,4] => [3,4,2,1] => 2
[2,3,1,1] => [3,1,2,4] => [2,4,3,1] => 2
[3,1,1,2] => [2,3,1,4] => [3,2,4,1] => 2
[3,1,2,1] => [2,3,1,4] => [3,2,4,1] => 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: St000507
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion 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 = 0 + 1
[1,1] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[1,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[1,1,1] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,1,2] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,2,1] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[2,1,1] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[1,1,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,3,1] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3,1,1] => [2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[1,2,2] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[2,1,2] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[2,2,1] => [3,1,2] => [[1,2],[3]]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [3,1,2] => [[1,2],[3]]
=> 2 = 1 + 1
[3,1,2] => [2,3,1] => [[1,3],[2]]
=> 2 = 1 + 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[1,1,1,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,1,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[2,1,1,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[1,1,1,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,3,1,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[3,1,1,1] => [2,3,1,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[1,1,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[1,4,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 3 = 2 + 1
[4,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 3 = 2 + 1
[1,1,2,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[2,1,1,2] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,1,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,2,1,1] => [3,1,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,3,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,1,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[1,3,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[1,3,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[2,1,1,3] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,1,3,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[2,3,1,1] => [3,1,2,4] => [[1,2,4],[3]]
=> 3 = 2 + 1
[3,1,1,2] => [2,3,1,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
[3,1,2,1] => [2,3,1,4] => [[1,3,4],[2]]
=> 3 = 2 + 1
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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ 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] => [1,2] => [2,1] => [2,1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,1,1] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[1,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[1,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[2,1,1] => [2,1,3] => [3,1,2] => [2,3,1] => 1
[1,1,3] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[1,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[3,1,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,2,2] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[2,1,2] => [2,1,3] => [3,1,2] => [2,3,1] => 1
[2,2,1] => [3,1,2] => [2,1,3] => [2,1,3] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [2,3,1] => 1
[2,3,1] => [3,1,2] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[2,1,1,1] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[3,1,1,1] => [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 2
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 2
[1,4,1,1] => [1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 2
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[2,1,1,2] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[2,1,2,1] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[2,2,1,1] => [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 2
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[2,1,1,3] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[2,1,3,1] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[2,3,1,1] => [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 2
[3,1,1,2] => [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 2
[3,1,2,1] => [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 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: St000024
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[2,1,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[2,1,1,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,3,1,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,1,1,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[2,1,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,3,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,3,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,1,1,3] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,2,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[2,1,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,1,1,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,3,1,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,1,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,1,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,2,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,3,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,3,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,1,1,3] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,3,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,3,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00069: Permutations —complement⟶ 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%
Mp00069: Permutations —complement⟶ 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] => [1,2] => [2,1] => [[1],[2]]
=> 1
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 1
[2,1] => [2,1] => [1,2] => [[1,2]]
=> 0
[1,1,1] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,1,2] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,2,1] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[2,1,1] => [2,1,3] => [2,3,1] => [[1,3],[2]]
=> 1
[1,1,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,3,1] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 1
[3,1,1] => [2,3,1] => [2,1,3] => [[1,3],[2]]
=> 1
[1,2,2] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[2,1,2] => [2,1,3] => [2,3,1] => [[1,3],[2]]
=> 1
[2,2,1] => [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,3,2] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [2,3,1] => [[1,3],[2]]
=> 1
[2,3,1] => [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [2,3,1] => [2,1,3] => [[1,3],[2]]
=> 1
[3,2,1] => [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,2,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,1,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[2,1,1,1] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,3,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,3,1,1] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[3,1,1,1] => [2,3,1,4] => [3,2,4,1] => [[1,4],[2],[3]]
=> 2
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,4,1] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[1,4,1,1] => [1,3,4,2] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,1,2] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,2,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[2,1,1,2] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[2,1,2,1] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[2,2,1,1] => [3,1,2,4] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,1,3,2] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,1,3] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,3,1] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,3,1,2] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,2,1] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[2,1,1,3] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[2,1,3,1] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[2,3,1,1] => [3,1,2,4] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[3,1,1,2] => [2,3,1,4] => [3,2,4,1] => [[1,4],[2],[3]]
=> 2
[3,1,2,1] => [2,3,1,4] => [3,2,4,1] => [[1,4],[2],[3]]
=> 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: St000168
Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [[]]
=> 0
[1,1] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> [[],[]]
=> 0
[1,1,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[1,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[1,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[2,1,1] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[1,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[1,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 1
[3,1,1] => [2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[1,2,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[2,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[2,2,1] => [3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[2,3,1] => [3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 0
[1,1,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,1,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,1,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,1,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[2,1,1,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[1,1,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,1,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,3,1,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[3,1,1,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[1,1,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,1,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,4,1,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[4,1,1,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[1,1,2,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[2,1,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,1,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,2,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[1,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,1,3,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,1,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,3,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,3,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[2,1,1,3] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,1,3,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,3,1,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[3,1,1,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[3,1,2,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
Description
The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. 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. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001581The achromatic number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St000299The number of nonisomorphic vertex-induced subtrees. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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. St000702The number of weak deficiencies of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St000259The diameter of a connected graph. St001820The size of the image of the pop stack sorting operator. St001626The number of maximal proper sublattices of a lattice. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000903The number of different parts of an integer composition. St001330The hat guessing number of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000933The number of multipartitions of sizes given by an integer partition. St001060The distinguishing index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000939The number of characters of the symmetric group whose value on the partition is positive. St001118The acyclic chromatic index of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001645The pebbling number of a connected graph. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000871The number of very big ascents of a permutation. St000035The number of left outer peaks of a permutation. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000422The energy of a graph, if it is integral. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000456The monochromatic index of a connected graph. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St000023The number of inner peaks of a permutation. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000682The Grundy value of Welter's game on a binary word. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001712The number of natural descents of a standard Young tableau. St001840The number of descents of a set partition. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001737The number of descents of type 2 in a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001637The number of (upper) dissectors of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
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