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Identifier
Values
=>
[1,2]=>2 [2,1]=>2 [1,2,3]=>6 [1,3,2]=>6 [2,1,3]=>6 [2,3,1]=>6 [3,1,2]=>6 [3,2,1]=>6 [1,2,3,4]=>24 [1,2,4,3]=>24 [1,3,2,4]=>24 [1,3,4,2]=>24 [1,4,2,3]=>24 [1,4,3,2]=>24 [2,1,3,4]=>24 [2,1,4,3]=>24 [2,3,1,4]=>24 [2,3,4,1]=>24 [2,4,1,3]=>25 [2,4,3,1]=>24 [3,1,2,4]=>24 [3,1,4,2]=>25 [3,2,1,4]=>24 [3,2,4,1]=>24 [3,4,1,2]=>24 [3,4,2,1]=>24 [4,1,2,3]=>24 [4,1,3,2]=>24 [4,2,1,3]=>24 [4,2,3,1]=>24 [4,3,1,2]=>24 [4,3,2,1]=>24 [1,2,3,4,5]=>120 [1,2,3,5,4]=>120 [1,2,4,3,5]=>120 [1,2,4,5,3]=>120 [1,2,5,3,4]=>120 [1,2,5,4,3]=>120 [1,3,2,4,5]=>120 [1,3,2,5,4]=>120 [1,3,4,2,5]=>120 [1,3,4,5,2]=>120 [1,3,5,2,4]=>125 [1,3,5,4,2]=>120 [1,4,2,3,5]=>120 [1,4,2,5,3]=>125 [1,4,3,2,5]=>120 [1,4,3,5,2]=>120 [1,4,5,2,3]=>120 [1,4,5,3,2]=>120 [1,5,2,3,4]=>120 [1,5,2,4,3]=>120 [1,5,3,2,4]=>120 [1,5,3,4,2]=>120 [1,5,4,2,3]=>120 [1,5,4,3,2]=>120 [2,1,3,4,5]=>120 [2,1,3,5,4]=>120 [2,1,4,3,5]=>120 [2,1,4,5,3]=>120 [2,1,5,3,4]=>120 [2,1,5,4,3]=>120 [2,3,1,4,5]=>120 [2,3,1,5,4]=>120 [2,3,4,1,5]=>120 [2,3,4,5,1]=>120 [2,3,5,1,4]=>126 [2,3,5,4,1]=>120 [2,4,1,3,5]=>125 [2,4,1,5,3]=>128 [2,4,3,1,5]=>120 [2,4,3,5,1]=>120 [2,4,5,1,3]=>126 [2,4,5,3,1]=>120 [2,5,1,3,4]=>126 [2,5,1,4,3]=>126 [2,5,3,1,4]=>121 [2,5,3,4,1]=>120 [2,5,4,1,3]=>126 [2,5,4,3,1]=>120 [3,1,2,4,5]=>120 [3,1,2,5,4]=>120 [3,1,4,2,5]=>125 [3,1,4,5,2]=>126 [3,1,5,2,4]=>128 [3,1,5,4,2]=>126 [3,2,1,4,5]=>120 [3,2,1,5,4]=>120 [3,2,4,1,5]=>120 [3,2,4,5,1]=>120 [3,2,5,1,4]=>126 [3,2,5,4,1]=>120 [3,4,1,2,5]=>120 [3,4,1,5,2]=>126 [3,4,2,1,5]=>120 [3,4,2,5,1]=>120 [3,4,5,1,2]=>120 [3,4,5,2,1]=>120 [3,5,1,2,4]=>126 [3,5,1,4,2]=>128 [3,5,2,1,4]=>126 [3,5,2,4,1]=>125 [3,5,4,1,2]=>120 [3,5,4,2,1]=>120 [4,1,2,3,5]=>120 [4,1,2,5,3]=>126 [4,1,3,2,5]=>120 [4,1,3,5,2]=>121 [4,1,5,2,3]=>126 [4,1,5,3,2]=>126 [4,2,1,3,5]=>120 [4,2,1,5,3]=>126 [4,2,3,1,5]=>120 [4,2,3,5,1]=>120 [4,2,5,1,3]=>128 [4,2,5,3,1]=>125 [4,3,1,2,5]=>120 [4,3,1,5,2]=>126 [4,3,2,1,5]=>120 [4,3,2,5,1]=>120 [4,3,5,1,2]=>120 [4,3,5,2,1]=>120 [4,5,1,2,3]=>120 [4,5,1,3,2]=>120 [4,5,2,1,3]=>120 [4,5,2,3,1]=>120 [4,5,3,1,2]=>120 [4,5,3,2,1]=>120 [5,1,2,3,4]=>120 [5,1,2,4,3]=>120 [5,1,3,2,4]=>120 [5,1,3,4,2]=>120 [5,1,4,2,3]=>120 [5,1,4,3,2]=>120 [5,2,1,3,4]=>120 [5,2,1,4,3]=>120 [5,2,3,1,4]=>120 [5,2,3,4,1]=>120 [5,2,4,1,3]=>125 [5,2,4,3,1]=>120 [5,3,1,2,4]=>120 [5,3,1,4,2]=>125 [5,3,2,1,4]=>120 [5,3,2,4,1]=>120 [5,3,4,1,2]=>120 [5,3,4,2,1]=>120 [5,4,1,2,3]=>120 [5,4,1,3,2]=>120 [5,4,2,1,3]=>120 [5,4,2,3,1]=>120 [5,4,3,1,2]=>120 [5,4,3,2,1]=>120
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Description
The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order.
Let $a(\pi)$ denote this statistic, then $a(\pi)$ is the product of [1] and [2]. A result of Sidorenko [3] implies that $a(\pi) \geq n!$ when $\pi$ is a permutation of $n$, with equality if and only if $\pi$ avoids the patterns $2413$ and $3142$. See [4] for a combinatorial proof and refinement. Upper bounds for $a(\pi)$ are given in [5].
References
[1] St000100oMp00065oMp00064
[2] St000100oMp00065
[3] Sidorenko, A. Inequalities for the number of linear extensions DOI:10.1007/BF00571183
[4] Gaetz, C., Gao, Y. Separable elements and splittings of Weyl groups arXiv:1911.11172
[5] Bollobás, Béla, Brightwell, G., Sidorenko, A. Geometrical Techniques for Estimating Numbers of Linear Extensions DOI:10.1006/eujc.1999.0299
Code
def statistic(pi):
    a1=pi.permutation_poset().linear_extensions().cardinality()
    a2=pi.reverse().permutation_poset().linear_extensions().cardinality()
    return a1*a2
Created
Jul 09, 2020 at 14:21 by Christian Gaetz
Updated
Jul 09, 2020 at 15:42 by Christian Stump