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Your data matches 265 different statistics following compositions of up to 3 maps.
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Matching statistic: St000156
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000156: 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] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => 3
[4,1,2,3] => [1,4,3,2] => 3
[4,2,1,3] => [1,4,3,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => 0
Description
The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
Matching statistic: St000004
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000004: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 3
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000019
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000019: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000030
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000030: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000030: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The sum of the descent differences of a permutations.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$$
See [[St000111]] and [[St000154]] for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the ''drop'' of a permutation.
Matching statistic: St000141
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000141: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000209
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000209: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000210
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000210: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
Minimum over maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the minimum of this value over all cycles in the permutation.
For example, all permutations with a fixed-point has statistic value 0,
and all permutations of $[n]$ with only one cycle, has statistic value $n-1$.
Matching statistic: St000305
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000305: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 3
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000316
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
St000316: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 2
[3,2,1] => [1,3,2] => [3,1,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 3
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St000334
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000334: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000334: 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] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 3
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The maz index, the major index of a permutation after replacing fixed points by zeros.
The descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1].
The following 255 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000868The aid statistic in the sense of Shareshian-Wachs. St001090The number of pop-stack-sorts needed to sort a permutation. St001375The pancake length of a permutation. St001541The Gini index of an integer partition. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001902The number of potential covers of a poset. St000054The first entry of the permutation. St000058The order of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000501The size of the first part in the decomposition of a permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000008The major index of the composition. St000009The charge of a standard tableau. St000018The number of inversions of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000051The size of the left subtree of a binary tree. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000171The degree of the graph. St000211The rank of the set partition. St000224The sorting index of a permutation. St000304The load of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000330The (standard) major index of a standard tableau. St000339The maf index of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000463The number of admissible inversions of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000651The maximal size of a rise in a permutation. St000692Babson and Steingrímsson's statistic of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001117The game chromatic index of a graph. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001434The number of negative sum pairs of a signed permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001696The natural major index of a standard Young tableau. St001721The degree of a binary word. St001726The number of visible inversions of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000013The height of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000110The number of permutations less than or equal to a permutation in left weak order. St000147The largest part of an integer partition. St000163The size of the orbit of the set partition under rotation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000240The number of indices that are not small excedances. St000321The number of integer partitions of n that are dominated by an integer partition. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000505The biggest entry in the block containing the 1. St000517The Kreweras number of an integer partition. St000655The length of the minimal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000734The last entry in the first row of a standard tableau. St000839The largest opener of a set partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000971The smallest closer of a set partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001110The 3-dynamic chromatic number of a graph. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001268The size of the largest ordinal summand in the poset. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001312Number of parabolic noncrossing partitions indexed by the composition. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001389The number of partitions of the same length below the given integer partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001481The minimal height of a peak of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001571The Cartan determinant of the integer partition. St001725The harmonious chromatic number of a graph. St001779The order of promotion on the set of linear extensions of a poset. St001809The index of the step at the first peak of maximal height in a Dyck path. St001838The number of nonempty primitive factors of a binary word. St000439The position of the first down step of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001814The number of partitions interlacing the given partition. St000653The last descent of a permutation. St000216The absolute length of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000815The number of semistandard Young tableaux of partition weight of given shape. St000844The size of the largest block in the direct sum decomposition of a permutation. St000391The sum of the positions of the ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000502The number of successions of a set partitions. St000503The maximal difference between two elements in a common block. St000539The number of odd inversions of a permutation. St000567The sum of the products of all pairs of parts. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000673The number of non-fixed points of a permutation. St000693The modular (standard) major index of a standard tableau. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000747A variant of the major index of a set partition. St000796The stat' of a permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000874The position of the last double rise in a Dyck path. St000877The depth of the binary word interpreted as a path. St000961The shifted major index of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001371The length of the longest Yamanouchi prefix of a binary word. St001480The number of simple summands of the module J^2/J^3. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001959The product of the heights of the peaks of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000422The energy of a graph, if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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 $eA$. 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 $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000219The number of occurrences of the pattern 231 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. 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. St001128The exponens consonantiae of a partition. St001684The reduced word complexity of a permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001555The order of a signed permutation. St001060The distinguishing index of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St000477The weight of a partition according to Alladi. 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. St000707The product of the factorials of the parts. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St001130The number of two successive successions in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000879The number of long braid edges in the graph of braid moves of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice.
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