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Identifier

Mp00248: Permutations —DEX composition⟶ Integer compositions

Images

[1] => [1]

[1,2] => [2]

[2,1] => [2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 13761 entries. <<<

click to show experimental identities (only identities of compositions of up to three maps are shown)

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.

References

[1] Shareshian, J., Wachs, M. L. Eulerian quasisymmetric functions arXiv:0812.0764

Properties

graded

Sage code

def mapping(pi):
    pi = list(pi)
    n = len(pi)
    for i, e in enumerate(pi):
        if e > i + 1:
            pi[i] = -n-1+e
    return Composition(from_subset=([i for i in range(1, len(pi)) if pi[i-1] > pi[i]], n))

Weight

Created

Feb 20, 2022 at 19:42 by Martin Rubey

Updated

Feb 20, 2022 at 19:42 by Martin Rubey