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Identifier

St001220: Permutations ⟶ ℤ

Values

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 872 entries. <<<

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Description

The width of a permutation.
Let $w$ be a permutation. The interval $[e,w]$ in the weak order is ranked, and we define $r_i=r_i(w)$ to be the number of elements at rank $i$ in $[e,w]$, where $i \in \{0, \dots, \ell(w)\}$. The width of
$w$ is the maximum of $\{r_0,r_1,\ldots,r_{\ell(w)}\}$. See [1].

References

[1] Fishel, S., Milićević, E., Patrias, R., Tenner, B. E. Enumerations relating braid and commutation classes arXiv:1708.04372

Code

def statistic(pi):
    S = SymmetricGroup(len(pi))
    pi = S(pi)
    P = S.weak_poset()
    return P.subposet(P.principal_order_ideal(pi)).width()

Created

Jun 26, 2018 at 12:54 by Susanna Fishel

Updated

Jun 26, 2018 at 13:13 by Christian Stump