Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000579
St000579: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 1
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 5
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 3
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 3
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 5
{{1},{2,4},{3}}
=> 4
{{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> 6
Description
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. This is the number of pairs $i\lt j$ in different blocks such that $j$ is the maximal element of a block.
Mp00080: Set partitions to permutationPermutations
St000156: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 1
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 3
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 2
{{1},{2,3}}
=> [1,3,2] => 2
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 6
{{1,2,3},{4}}
=> [2,3,1,4] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => 4
{{1,2},{3,4}}
=> [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => 5
{{1,3},{2,4}}
=> [3,4,1,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => 5
{{1},{2,3,4}}
=> [1,3,4,2] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => 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]].
Mp00080: Set partitions to permutationPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
St000339: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 3
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 4
{{1,2,4},{3}}
=> [2,4,3,1] => [3,2,4,1] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 5
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => 6
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 5
{{1,4},{2,3}}
=> [4,3,2,1] => [2,3,4,1] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [3,4,2,1] => 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The maf index of a permutation. Let $\sigma$ be a permutation with fixed point set $\operatorname{FIX}(\sigma)$, and let $\operatorname{Der}(\sigma)$ be the derangement obtained from $\sigma$ by removing the fixed points. Then $$\operatorname{maf}(\sigma) = \sum_{i \in \operatorname{FIX}(\sigma)} i - \binom{|\operatorname{FIX}(\sigma)|+1}{2} + \operatorname{maj}(\operatorname{Der}(\sigma)),$$ where $\operatorname{maj}(\operatorname{Der}(\sigma))$ is the major index of the derangement of $\sigma$.
Mp00080: Set partitions to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000446: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 6
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 5
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 4
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 5
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The disorder of a permutation. Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process. For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Mp00080: Set partitions to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
St000794: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 6
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 4
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 5
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 5
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The mak of a permutation. According to [1], this is the sum of the number of occurrences of the vincular patterns $(2\underline{31})$, $(\underline{32}1)$, $(1\underline{32})$, $(\underline{21})$, where matches of the underlined letters must be adjacent.
Mp00080: Set partitions to permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000833: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
=> [3,2,1] => [3,1,2] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 5
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,1,3] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => 6
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,3,1] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,1,2,4] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [4,1,2,3] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,1,2] => 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The comajor index of a permutation. This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Mp00080: Set partitions to permutationPermutations
Mp00089: Permutations Inverse Kreweras complementPermutations
St001759: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => 3
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => 5
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => 4
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => 6
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => 5
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => 3
Description
The Rajchgot index of a permutation. The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
Mp00080: Set partitions to permutationPermutations
Mp00086: Permutations first fundamental transformationPermutations
Mp00241: Permutations invert Laguerre heapPermutations
St000004: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,1,2] => [2,3,1] => 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => [3,1,4,2] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,1,3] => [3,4,2,1] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 6
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,3,1] => [4,3,1,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,1,2,4] => [2,3,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,1,2,3] => [2,3,4,1] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,1,2] => [2,4,3,1] => 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 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'''.
Mp00080: Set partitions to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [[1],[2]]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [[1,2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [[1,3],[2]]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 5
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => [[1,4],[2],[3]]
=> 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [[1,3],[2],[4]]
=> 4
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [[1,3,4],[2]]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [[1,4],[2],[3]]
=> 5
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
Description
The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.
Mp00080: Set partitions to permutationPermutations
Mp00236: Permutations Clarke-Steingrimsson-Zeng inversePermutations
Mp00073: Permutations major-index to inversion-number bijectionPermutations
St000224: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [3,1,2] => 3
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [2,3,1] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => [4,3,1,2] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [4,2,3,1] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 5
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [3,4,2,1] => 5
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [4,1,2,3] => 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [2,4,1,3] => 4
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
Description
The sorting index of a permutation. The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols. Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000246The number of non-inversions of a permutation. St000304The load of a permutation. St000305The inverse major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000391The sum of the positions of the ones in a binary word. St000692Babson and Steingrímsson's statistic of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St001161The major index north count of a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected 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. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.