- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤ

Values

[[1],[2],[3]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[2],[4]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[3],[4]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[3],[4]] => [3,2,1] => [2,1] => [1,1] => 3

[[1,1],[2],[3]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,2],[2],[3]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,3],[2],[3]] => [3,2,1,4] => [2,2] => [2] => 3

[[1],[2],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[3],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[4],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[3],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[4],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[3],[4],[5]] => [3,2,1] => [2,1] => [1,1] => 3

[[1,1],[2],[4]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,1],[3],[4]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,2],[2],[4]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,2],[3],[4]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,3],[2],[4]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,4],[2],[3]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,4],[2],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,3],[3],[4]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,4],[3],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[2,2],[3],[4]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[2,3],[3],[4]] => [4,2,1,3] => [2,2] => [2] => 3

[[2,4],[3],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[1],[2],[3],[4]] => [4,3,2,1] => [1,2,1] => [1,1,1] => 4

[[1,1,1],[2],[3]] => [5,4,1,2,3] => [1,4] => [1,1] => 3

[[1,1,2],[2],[3]] => [5,3,1,2,4] => [1,4] => [1,1] => 3

[[1,1,3],[2],[3]] => [4,3,1,2,5] => [1,4] => [1,1] => 3

[[1,2,2],[2],[3]] => [5,2,1,3,4] => [2,3] => [1,1] => 3

[[1,2,3],[2],[3]] => [4,2,1,3,5] => [2,3] => [1,1] => 3

[[1,3,3],[2],[3]] => [3,2,1,4,5] => [2,3] => [1,1] => 3

[[1,1],[2,2],[3]] => [5,3,4,1,2] => [1,4] => [1,1] => 3

[[1,1],[2,3],[3]] => [4,3,5,1,2] => [1,4] => [1,1] => 3

[[1,2],[2,3],[3]] => [4,2,5,1,3] => [2,3] => [1,1] => 3

[[1],[2],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[3],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[4],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[1],[5],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[3],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[4],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[2],[5],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[3],[4],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[3],[5],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[4],[5],[6]] => [3,2,1] => [2,1] => [1,1] => 3

[[1,1],[2],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,1],[3],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,1],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,2],[2],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,2],[3],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,3],[2],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,5],[2],[3]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,2],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,4],[2],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,5],[2],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,5],[2],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,3],[3],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,3],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[1,4],[3],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,5],[3],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,5],[3],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[1,4],[4],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[1,5],[4],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[2,2],[3],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[2,2],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[2,3],[3],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[2,3],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[2,4],[3],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[2,5],[3],[4]] => [3,2,1,4] => [2,2] => [2] => 3

[[2,5],[3],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[2,4],[4],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[2,5],[4],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[3,3],[4],[5]] => [4,3,1,2] => [1,3] => [1,1] => 3

[[3,4],[4],[5]] => [4,2,1,3] => [2,2] => [2] => 3

[[3,5],[4],[5]] => [3,2,1,4] => [2,2] => [2] => 3

[[1],[2],[3],[5]] => [4,3,2,1] => [1,2,1] => [1,1,1] => 4

[[1],[2],[4],[5]] => [4,3,2,1] => [1,2,1] => [1,1,1] => 4

[[1],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => [1,1,1] => 4

[[2],[3],[4],[5]] => [4,3,2,1] => [1,2,1] => [1,1,1] => 4

[[1,1,1],[2],[4]] => [5,4,1,2,3] => [1,4] => [1,1] => 3

[[1,1,1],[3],[4]] => [5,4,1,2,3] => [1,4] => [1,1] => 3

[[1,1,2],[2],[4]] => [5,3,1,2,4] => [1,4] => [1,1] => 3

[[1,1,2],[3],[4]] => [5,4,1,2,3] => [1,4] => [1,1] => 3

[[1,1,3],[2],[4]] => [5,3,1,2,4] => [1,4] => [1,1] => 3

[[1,1,4],[2],[3]] => [4,3,1,2,5] => [1,4] => [1,1] => 3

[[1,1,4],[2],[4]] => [4,3,1,2,5] => [1,4] => [1,1] => 3

[[1,1,3],[3],[4]] => [5,3,1,2,4] => [1,4] => [1,1] => 3

[[1,1,4],[3],[4]] => [4,3,1,2,5] => [1,4] => [1,1] => 3

[[1,2,2],[2],[4]] => [5,2,1,3,4] => [2,3] => [1,1] => 3

[[1,2,2],[3],[4]] => [5,4,1,2,3] => [1,4] => [1,1] => 3

[[1,2,3],[2],[4]] => [5,2,1,3,4] => [2,3] => [1,1] => 3

[[1,2,4],[2],[3]] => [4,2,1,3,5] => [2,3] => [1,1] => 3

[[1,2,4],[2],[4]] => [4,2,1,3,5] => [2,3] => [1,1] => 3

[[1,2,3],[3],[4]] => [5,3,1,2,4] => [1,4] => [1,1] => 3

[[1,3,3],[2],[4]] => [5,2,1,3,4] => [2,3] => [1,1] => 3

[[1,3,4],[2],[3]] => [3,2,1,4,5] => [2,3] => [1,1] => 3

[[1,2,4],[3],[4]] => [4,3,1,2,5] => [1,4] => [1,1] => 3

[[1,3,4],[2],[4]] => [4,2,1,3,5] => [2,3] => [1,1] => 3

[[1,4,4],[2],[3]] => [3,2,1,4,5] => [2,3] => [1,1] => 3

[[1,4,4],[2],[4]] => [3,2,1,4,5] => [2,3] => [1,1] => 3

[[1,3,3],[3],[4]] => [5,2,1,3,4] => [2,3] => [1,1] => 3

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search for individual values

searching the database for the individual values of this statistic

Description

The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.

Map

DEX composition

Description

The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.