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St001841: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
Description
The number of inversions of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. A pair $(i,j)$ is an inversion of the word $w$ if $w_i > w_j$.
Mp00215: Set partitions Wachs-WhiteSet partitions
St000572: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
{{1,3},{2,4}}
=> {{1,3,4},{2}}
=> 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
Description
The dimension exponent of a set partition. This is $$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$ where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$. It is thus equal to the difference [[St000728]] - [[St000211]]. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block. This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
Mp00115: Set partitions Kasraoui-ZengSet partitions
St001843: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
Description
The Z-index of a set partition. The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$. The Z-index of $w$ equals $$ \sum_{i < j} w_{i,j}, $$ where $w_{i,j}$ is the word obtained from $w$ by removing all letters different from $i$ and $j$.
Mp00217: Set partitions Wachs-White-rho Set partitions
Mp00171: Set partitions intertwining number to dual major indexSet partitions
St000586: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal.
Mp00218: Set partitions inverse Wachs-White-rhoSet partitions
Mp00216: Set partitions inverse Wachs-WhiteSet partitions
St000610: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,2,4},{3}}
=> 2
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal.
Mp00112: Set partitions complementSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Mp00112: Set partitions complementSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000019: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 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.
Mp00112: Set partitions complementSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000029: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The depth of a permutation. This is given by $$\operatorname{dp}(\sigma) = \sum_{\sigma_i>i} (\sigma_i-i) = |\{ i \leq j : \sigma_i > j\}|.$$ The depth is half of the total displacement [4], Problem 5.1.1.28, or Spearman’s disarray [3] $\sum_i |\sigma_i-i|$. Permutations with depth at most $1$ are called ''almost-increasing'' in [5].
Mp00112: Set partitions complementSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000030: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [1,3,2] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 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: St000034
Mp00215: Set partitions Wachs-WhiteSet partitions
Mp00080: Set partitions to permutationPermutations
Mp00277: Permutations catalanizationPermutations
St000034: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => 0
{{1,2,4},{3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [4,3,2,1] => 2
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => 1
{{1,3},{2,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 1
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 2
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [3,4,2,1] => 2
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000463The number of admissible inversions of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000670The reversal length of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001090The number of pop-stack-sorts needed to sort a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001569The maximal modular displacement of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001684The reduced word complexity of a permutation. St001842The major index of a set partition. St000058The order of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. 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. St001857The number of edges in the reduced word graph of a signed permutation. St001623The number of doubly irreducible elements of a lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.